Differentiate:

To find: Differentiation of $x^{n} \cot x$

Formula used: (i) (uv)' = u'v + uv' (Leibnitz or product rule)

(ii)

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

(iii)

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

Let us take $u=x^{n}$ and $v=\cot x$

$u^{\prime}=\frac{d u}{d x}=\frac{d x^{n}}{d x}=n x^{n-1}$

$v^{\prime}=\frac{d v}{d x}=\frac{d \cot x}{d x}=-\operatorname{cosec}^{2} x$

Putting the above obtained values in the formula :-

$(u v)^{\prime}=u^{\prime} v+u v^{\prime}$

$\left(x^{n} \cot x\right)^{\prime}=n x^{n-1} \times \cot x+x^{n} x-\operatorname{cosec}^{2} x$

$=n x^{n-1} \cot x-x^{n} \operatorname{cosec}^{2} x$

$=x^{n}\left(n x^{-1} \cot x-\operatorname{cosec}^{2} x\right)$

Ans) $x^{n}\left(n x^{-1} \cot x-\operatorname{cosec}^{2} x\right)$