Question

Find $\frac{dy}{dx}$, if $y={\cos }^{-1}\left(\sin 5x\right)$

Answer 1

$y={\cos }^{-1}\left(\sin 5\left(x\right)\right)$

Or $\cos y=\sin \left(5x\right)$

Taking derivative of both sides,

$-\sin y\frac{dy}{dx}=\cos \left(5x\right).5$

Or $\frac{dy}{dx}=\frac{5\cos 5\left(x\right)}{-\sin y}$

Or $\frac{dy}{dx}=\frac{-5\cos 5\left(x\right)}{\sin \left({\cos }^{-1}\left(\sin \left(5x\right)\right)\right)}$