Differentiate:

To find: Differentiation of $x^{4} \tan x$

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

(ii)

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

(iii)

$\frac{d \tan x}{d x}=\sec ^{2} x$

Let us take $u=x^{4}$ and $v=\tan x$

$u^{\prime}=\frac{d u}{d x}=\frac{d x^{4}}{d x}=4 x^{3}$

$v^{\prime}=\frac{d v}{d x}=\frac{d \tan x}{d x}=\sec ^{2} x$

Putting the above obtained values in the formula:-

$(U V)^{\prime}=U^{\prime} V+U V^{\prime}$

$\left(x^{4} \tan x\right)^{\prime}=4 x^{3} \times \tan x+x^{4} \times \sec ^{2} x$

$=4 x^{3} \tan x+x^{4} \sec ^{2} x$

$=x^{3}\left(4 \tan x+x \sec ^{2} x\right)$

Ans) $x^{3}\left(4 \tan x+x \sec ^{2} x\right)$