Question

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

Answer 1

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

On differentiating w.r.t $x$, we get

$\frac{dy}{dx}=\frac{d}{dx}\left[\sin \right(2{\cos }^{-1}x\left)\right]$

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

$\frac{dy}{dx}=\left[\cos \right(2{\cos }^{-1}x\left)\right]\times 2\times \frac{-1}{\sqrt{1-x^2}}$

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

Hence, this is the answer.