Question

Find the derivatives of the following functions.
(a) ${\cot }^{3}x$ (b) $\sin \sqrt{x}$

Answer 1

$\left(a\right)\frac{d}{dx}{\cot }^{3}x$

Let, $z=\cot x⟹\frac{dz}{dx}=-{\text{cosec}}^{2}x$

$=\frac{d{z}^3}{dx}=\frac{d{z}^3}{dz}\frac{dz}{dx}$

$=-3z^2{\text{cosec}}^{2}x$

$=-3{\cot }^{2}x{\text{cosec}}^{2}x$

$\left(b\right)\frac{d}{dx}\sin \sqrt{x}$

Let $z=\sqrt{x}⟹\frac{dz}{dx}=\frac{1}{2\sqrt{x}}$

$=\frac{d\sin z}{dx}=\frac{d\sin z}{dz}\frac{dz}{dx}$

$=\cos z\frac{1}{2\sqrt{x}}$

$=\cos \sqrt{x}\frac{1}{2\sqrt{x}}$