Question

Find the derivative of $y=\sin \left({\cos }^2\right({\tan }^{3}x\left)\right)$

Answer 1

$y=\sin \left({\cos }^2\right({\tan }^{3}x\left)\right)$

By applying chain rule of differentiation:

$\frac{dy}{dx}=\frac{d}{dx}\left(\sin x\right({\cos }^2{\tan }^{3}x\left)\right)$

$=\cos \left({\cos }^2\right({\tan }^{3}x\left)\right)\cdot 2\cos \left({\tan }^{3}x\right)-\sin \left({\tan }^{3}x\right)\cdot 3{\tan }^{2}x\cdot {\sec }^{2}x$

$=\frac{-3{\tan }^{2}x}{{\cos }^{2}x}\sin \left({\tan }^{3}x\right)\cdot 2\cos \left({\tan }^{3}x\right)\cdot \cos \left({\cos }^2\right({\tan }^{3}x\left)\right)$.