Question

Differentiate following functions w.r.t x
(i) $2\sqrt{\cot \left(x^2\right)}$ (ii) $\cos \left(\sqrt{x}\right)$

Answer 1

$\left(i\right)$

Let $y=cotx$

$=>y=\frac{cosx}{sinx}$

let $u=cosx$ and $v=sinx$

now,differentiate w.r.t $x$

$=>\frac{dy}{dx}=\frac{cosx}{sinx}$

Now this is in the form of $y=\frac{u}{v}$

then,,

$y^{\prime }=\frac{u^{\prime }v-v^{\prime }u}{v^2}$

now,

$=>y^{\prime }=\frac{-sinx\times sinx-cosx\times cosx}{si{n}^x}$

$=>\frac{-(si{n}^{2}x+co{s}^{2}x)}{sin{x}^2}$

$=>\frac{-1}{s{n}^{2}x}$

$=>-cose{c}^{2}x$

differential of $cotx$ is $-cose{c}^x$

now,

$2\sqrt{-}cose{c}^{2}x=2(-cosecx)$

$=>-2coscex$

$(ii.)$$

$=>y=cos\sqrt{x}$

$\frac{dy}{dx}=cos\sqrt{x}$

$=>-sin\sqrt{x}\times \frac{1}{2\sqrt{x}}$