Question

Find the derivative of the following functions form first principle:
$(-x){}^{-1}$

Answer 1

Let $f\left(x\right)=(-x){}^{-1}=\frac{-1}{x}$
Thus using first principle
$f^{\prime }\left(x\right)=\underset{x\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
= $\underset{x\to 0}{lim}\frac{1}{h}[\frac{-1}{x+h}-\left(\frac{-1}{x}\right)]$
= $\underset{x\to 0}{lim}\frac{1}{h}[\frac{-1}{x+h}+\frac{1}{x}]$
=$\underset{x\to 0}{lim}\frac{1}{h}\left[\frac{-x+(x+h)}{x(x+h)}\right]$
$=\underset{h\to 0}{lim}\frac{1}{h}\left[\frac{-x+x+h}{x(x+h)}\right]$
$=\underset{x\to 0}{lim}\frac{1}{h}\left[\frac{h}{x(x+h)}\right]$
$=\underset{x\to 0}{lim}\frac{1}{x(x+h)}=\frac{1}{x^2}$