Question

Solve
$\frac{d}{dx}\frac{1}{\sqrt{x^2-a^2}}$

Answer 1

We have,

$\frac{d}{dx}\frac{1}{\sqrt{x^2-a^2}}$

$\frac{d}{dx}\frac{1}{{(x^2-a^2)}^{1/2}}$

$\frac{d}{dx}{(x^2-a^2)}^{1/2}$

we know that,

$\Rightarrow \frac{d}{dx}{x}^n=n{x}^{n-1}$

$\Rightarrow -\frac{1}{2}{(x^2-a^2)}^{-\frac{1}{2}-1}\frac{d}{dx}(x^2-a^2)$

$\Rightarrow -\frac{1}{2}{(x^2-a^2)}^{-\frac{3}{2}}(2x-0)$

$\Rightarrow -\frac{1}{2}\frac{1}{{(x^2-a^2)}^{-\frac{3}{2}}}2x$

$\Rightarrow -\frac{1}{(x^2-a^2)\sqrt{x^2-a^2}}$

Hence, this is the answer.