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

The definition of a derivative is:

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

Step 1: Write down the definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ).

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

Step 3: Simplify the expression: ( f'(x) = \lim_{h \to 0} \frac{\frac{x - (x + h)}{x(x + h)}}{h} ).

Step 4: Further simplify the expression: ( f'(x) = \lim_{h \to 0} \frac{-1}{x(x + h)h} ).

Step 5: Evaluate the limit as ( h ) approaches 0: ( f'(x) = \frac{-1}{x^2} ).

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

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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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