Question

Differentiate :
$\frac{d}{dx}\left({\sin }^{-1}x\right)$

Answer 1

$\frac{d\left({\sin }^{-1}x\right)}{dx}$

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

$\sin y=x$

$\cos y\frac{dy}{dx}=1$

$\frac{dy}{dx}=\sec y=\sec \left({\sin }^{-1}x\right)$

from triangle

${\sin }^{-1}x={\sec }^{-1}\frac{1}{\sqrt{1-x^2}}$

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