How do you find f'(x) using the definition of a derivative #f(x) =x^2 - 1#?

Question

Emma Aldridge · Accepted Answer

#f'(x)=2x# The definition of a derivative is: #f'(x)=lim_(hto0)(f(x+h)-f(x))/h# Here, #f(x)=x^2-1# #f'(x)=lim_(hto0)((x+h)^2-1-x^2+1)/h# #f'(x)=lim_(hto0)(x^2+2xh+h^2-x^2)/h# #f'(x)=lim_(hto0)(2xh+h^2)/h# #f'(x)=lim_(hto0)2x+h=2x+0=2x# #f'(x)=2x# for #f(x)=x^2-1#

Emily Ammerman · Answer

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To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these stepsTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

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To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1.To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x)To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) =To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{fTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \limTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x +To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h)To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) -To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - fTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{fTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x +To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

SubTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

SubstituteTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h)To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute theTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) -To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the functionTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - fTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ fTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x)To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) =To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 -To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2.To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. SubstituteTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute theTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 \To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the functionTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ intoTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ fTo find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into theTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(xTo find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formulaTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x)To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) =To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = xTo find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ fTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 -To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x)To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 \To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \limTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ intoTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into theTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definitionTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ fTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x +To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \limTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 -To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 -To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x +To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 -To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 -To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

ExpandTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 -To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x +To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + hTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 \To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ andTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplifyTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x)To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (xTo find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x +To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + hTo find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \limTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ andTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplifyTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ fTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x)To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 +To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xhTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh +To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 -To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 -To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 +To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xhTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh +To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ fTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 -To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ fTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \fracTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ fTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xhTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \limTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4.To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. FactorTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor outTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x +To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + hTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h \To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h)To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from theTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now,To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substituteTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ hTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h =To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x)To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \limTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ intoTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into theTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ fTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2xTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x \To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

SoTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So,To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, theTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivativeTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative ofTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of theTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ fTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5.To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. CancelTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x)To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel outTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ hTo find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h \To find \( f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from theTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numeratorTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator andTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ withTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect toTo find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ fTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(xTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x \To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x)To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) =To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ fTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \limTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{hTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(x)To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(x) =To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \toTo find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(x) = To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \to To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(x) = 2To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \to 0To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(x) = 2xTo find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \to 0}To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(x) = 2x \To find \( f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \to 0} (2To find $ f'(x) $ using the definition of a derivative for the function $ f(x) = x^2 - 1 $, we use the formula:

\[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

Substitute the function $ f(x) = x^2 - 1 $ into the formula:

$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

Expand $ (x + h)^2 $ and simplify:

$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

Now, substitute $ h = 0 $ into the expression:

$$ f'(x) = 2x $$

So, the derivative of the function $ f(x) = x^2 - 1 $ with respect to $ x $ is $ f'(x) = 2x $.To find $ f'(x) $ using the definition of a derivative for $ f(x) = x^2 - 1 $, follow these steps:

1. Start with the definition of the derivative:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Substitute the function $ f(x) = x^2 - 1 $ into the definition:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - 1 - (x^2 - 1)}{h} $$

3. Expand $ (x + h)^2 $ and simplify:
$$ f'(x) = \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - 1 - x^2 + 1}{h} $$
$$ f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} $$

4. Factor out $ h $ from the numerator:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$

5. Cancel out $ h $ from the numerator and denominator:
$$ f'(x) = \lim_{h \to 0} (2x + h) $$

6. Evaluate the limit as $ h $ approaches 0:
$$ f'(x) = 2x $$

Therefore, the derivative of $ f(x) = x^2 - 1 $ is $ f'(x) = 2x $.