4.(iii)   Find the derivative of the following functions from first principle. 1 / x ^2

Answers (1)

f(x)= 1 / x ^2

Now, As we know, The derivative of any function at x is 

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

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

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

f'(x)=\lim_{h\rightarrow 0} \frac{x^2-x^2-2xh-h^2}{h(x+h)^2x^2}

f'(x)=\lim_{h\rightarrow 0} \frac{-2xh-h^2}{h(x+h)^2x^2}

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

f'(x)= \frac{-2x-0}{(x+0)^2x^2}

f'(x)= \frac{-2}{x^3}  (Answer)

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