Differentiate:

Question:

Differentiate:

$x^{2} \sin x$

 

Solution:

To find: Differentiation of $x^{2} \sin x$

Formula used: (i) (uv) $^{\prime}=u^{\prime} v+u v^{\prime}$ (Leibnitz or product rule)

(ii)

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

(iii)

$\frac{d \sin x}{d x}=\cos x$

Let us take $u=x^{2}$ and $v=\sin x$

$u^{\prime}=\frac{d u}{d x}=\frac{d\left(x^{2}\right)}{d x}=2 x$

$v^{\prime}=\frac{d v}{d x}=\frac{d(\sin x)}{d x}=\cos x$

Putting the above obtained values in the formula:-

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

$\left(x^{2} \sin x\right)^{\prime}=2 x \times \sin x+x^{2} \times \cos x$

$=2 x \sin x+x^{2} \cos x$

Ans) $2 x \sin x+x^{2} \cos x$

 

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