Differentiate

To find: Differentiation of $\frac{2 \cot x}{\sqrt{x}}$

Formula used: (i) $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule)

(ii) $\frac{d \cot x}{d x}=-\operatorname{cosec}^{2} x$

(iii) $\frac{d x^{n}}{d x}=n x^{n-1}$

Let us take $u=(2 \cot x)$ and $v=$

$(\sqrt{x})$

$u^{\prime}=\frac{d u}{d x}=\frac{d(2 \cot x)}{d x}=-2 \operatorname{cosec}^{2} x$

$v^{\prime}=\frac{d v}{d x}=\frac{d(\sqrt{x})}{d x}=\frac{1}{2 \sqrt{x}}$

Putting the above obtained values in the formula:-

$\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule)

$\left[\frac{2 \cot x}{\sqrt{x}}\right]^{\prime}=\frac{-2 \operatorname{cosec}^{2} x \times(\sqrt{x})-(2 \cot x) \times\left(\frac{1}{2 \sqrt{x}}\right)}{(\sqrt{x})^{2}}$

$=\frac{-2 x \operatorname{cosec}^{2} x-(\cot x)}{\sqrt{x}(\sqrt{x})^{2}}$

Ans $)=\frac{-2 x \operatorname{cosec}^{2} x-\cot x}{x^{3 / 2}}$