Differentiate
$\frac{x^{4}}{\sin x}$
To find: Differentiation of $\frac{x^{4}}{\sin x}$
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) $\frac{d x^{n}}{d x}=n x^{n-1}$
(iii) $\frac{d \sin x}{d x}=\cos x$
Let us take $u=\left(x^{4}\right)$ and $v=(\sin x)$
$u^{\prime}=\frac{d u}{d x}=\frac{d\left(x^{4}\right)}{d x}=4 x^{3}$
$v^{\prime}=\frac{d v}{d x}=\frac{d(\sin x)}{d x}=\cos x$
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)
$\left[\frac{x^{4}}{\sin x}\right]=\frac{4 x^{3} \times(\sin x)-\left(x^{4}\right) \times(\cos x)}{(\sin x)^{2}}$
$=\frac{x^{3}[4(\sin x)-x(\cos x)]}{(\sin x)^{2}}$
Ans $)=\frac{x^{3}[4(\sin x)-x(\cos x)]}{(\sin x)^{2}}$
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