Question

Differentiate the following functions w.r.t. $x$ :
${\sin }^{-1}\left(1-x^2\right)$

Answer 1

Let $y={\sin }^{-1}(1-x^2)$

Differentiate on both sides w.r.t. x

$\frac{dy}{dx}=\frac{1}{\sqrt{1-{(1-x^2)}^2}}\frac{d}{dx}(1-x^2)$

$=\frac{-2x}{\sqrt{2x^2-x^4}}-1\le 1-x^2\le 1;2\ge x^2\ge 0;xϵ[-\sqrt{2}\le x\le \sqrt{2}]$

$\therefore \frac{dy}{dx}=\frac{-2x}{\left|x\right|\sqrt{2-x^2}},(-\sqrt{2}\le x\le \sqrt{2})$

$\therefore \frac{dy}{dx}=\{\begin{array}{c}\frac{2}{\sqrt{2-x^2}},-\sqrt{2}\le x\le 0\\ \frac{-2}{\sqrt{2-x^2}},0<x\le \sqrt{2}\end{array}$