How do you use the definition of a derivative to find the derivative of #f(x) =1/x#?

Answer 1

When you are asked to differentiate using the definition of the derivative, you need to use the formula #f'(x) =lim_(h -> 0) (f(x + h) - f(x))/h#

#f'(x) = lim_(h -> 0) (1/(x + h) - 1/x)/h#
#f'(x) = lim_(h -> 0) (x/(x(x + h)) - (x + h)/(x(x + h)))/h#
#f'(x) = lim_(h-> 0) (-h/(x(x + h))) xx 1/h#
#f'(x) = lim_(h -> 0) -1/(x^2 + xh)#
#f'(x) = -1/(x^2 + x xx 0)#
#f'(x) = -1/x^2#

Using the power rule to confirm:

#f(x) = x^-1#
#f'(x) = -1x^(-1 - 1)#
#f'(x) = -1x^(-2)#
#f'(x) = -1/x^2#

Hopefully this helps!

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Answer 2

To find the derivative of (f(x) = \frac{1}{x}) using the definition of a derivative, we start with the definition:

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

Substitute (f(x) = \frac{1}{x}) into the definition:

[f'(x) = \lim_{h \to 0} \frac{\frac{1}{x + h} - \frac{1}{x}}{h}]

Combine the fractions in the numerator:

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

Simplify the numerator:

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

Cancel out the (h) in the numerator and denominator:

[f'(x) = \lim_{h \to 0} \frac{-1}{x(x + h)}]

Take the limit as (h) approaches 0:

[f'(x) = \frac{-1}{x^2}]

So, the derivative of (f(x) = \frac{1}{x}) is (f'(x) = \frac{-1}{x^2}).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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