Browse Questions

# Find the derivative of the given function from first principle  $\Large\frac{1}{x^2}$.

Let $f(x) = \large\frac{1}{x^2}$. Accordingly, from the first principle.
$f'(x) \lim\limits_{ h \to 0}\large\frac{f(x+h)-f(x)}{h}$
$\lim\limits_{ h \to 0}\large\frac{\Large\frac{1}{(x+h)^2}-\Large\frac{1}{x^2}}{h}$
$= \lim\limits_{h \to 0} \large\frac{1}{h}$$\bigg[ \large\frac{x^2-(x+h)^2}{x^2(x+h)^2} \bigg] = \lim\limits_{h \to 0} \large\frac{1}{h}$$\bigg[ \large\frac{x^2-x^2-h^2-2hx}{x^2(x+h)^2} \bigg]$
$= \lim\limits_{h \to 0} \large\frac{1}{h}$$\bigg[ \large\frac{-h^2-2hx}{x^2(x+h)^2} \bigg] = \lim\limits_{h \to 0} \large\frac{1}{h}$$\bigg[ \large\frac{-h-2x}{x^2(x+h)^2} \bigg]$
$= \large\frac{0-2x}{x^2(x+0)^2}$$\large\frac{-2}{x^3}$