Question

Differentiate the given function w.r.t. $x$:
${\sin }^{-1}\left(x\sqrt{x}\right),0\le x\le 1$

Answer 1

Let $y={\sin }^{-1}\left(x\sqrt{x}\right)$
Using chain rule, we obtain
$\frac{dy}{dx}=\frac{d}{dx}{\sin }^{-1}\left(x\sqrt{x}\right)$
$=\frac{1}{\sqrt{1-(x\sqrt{x}{)}^2}}\times \frac{d}{dx}\left(x\sqrt{x}\right)$
$=\frac{1}{\sqrt{1-x^3}}.\frac{d}{dx}\left(x^{\frac{3}{2}}\right)$
$=\frac{1}{\sqrt{1-x^3}}\times \frac{3}{2}.x^{\frac{1}{2}}$
$=\frac{3\sqrt{x}}{2\sqrt{1-x^3}}$
$=\frac{3}{2}\sqrt{\frac{x}{1-x^3}}$