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Using the definition of the derivative, find the derivative of: f(x)= \frac{1}{x^2} at x=a.

Question:

Using the definition of the derivative, find the derivative of:

{eq}f(x)= \frac{1}{x^2} {/eq}

at x=a.

Answer and Explanation:

{eq}f(x)= \frac{1}{x^2} \\ \displaystyle f'(a) =\lim _{h\rightarrow0}\frac{f(a+h) - f(a) }{h}\\ \displaystyle f(a+h) = \frac{1}{(a+h)^2}\\ \displaystyle f'(a) =\lim _{h\rightarrow0}\frac{ \frac{1}{(a+h)^2} - \frac{1}{(a)^2} }{h}\\ {/eq}

{eq}\displaystyle f'(a) =\lim _{h\rightarrow0}\frac{ \frac{1}{(a+h)^2} - \frac{1}{(a)^2} }{h}\\ \displaystyle f'(a) =\lim _{h\rightarrow0}\frac{a^2-(a+h)^2}{ha^2(a+h)^2}\\ \displaystyle f'(a) =\lim _{h\rightarrow0}\frac{-2a-h}{a^2(a+h)^2}\\ \displaystyle f'(a) = \frac{-2}{a^3} {/eq}


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