Question

Differentiate with respect to $x$:
$x^{{\sin }^{-1}x}$

Answer 1

$y=x^{{\sin }^{-1}x}$

taking log on both sides

$\log y={\sin }^{-1}x\log x$

$\frac{1}{y}\frac{dy}{dx}=\frac{{\sin }^{-1}x}{x}+\frac{\log x}{\sqrt{1-x^2}}$

$\frac{dy}{dx}=x^{{\sin }^{-1}x}[\frac{{\sin }^{-1}x}{x}+\frac{\log x}{\sqrt{1-x^2}}]$