Question

Find the derivative of $si{n}^{2}x$ with respect to x using product rule.

Answer 1

$\frac{d}{dx}\left({\sin }^{2}x\right)$

$=\frac{d}{dx}(\sin x.\sin x)$

Applying product rule, we have

$\frac{d}{dx}(\sin x.\sin x)$

$=\sin x\frac{d}{dx}\left(\sin x\right)+\frac{d}{dx}\left(\sin x\right)\sin x$

$=\sin \cos x+\cos x\sin x(\because \frac{d}{dx}\left(\sin x\right)=\cos x)$

$=2\sin x\cos x$

$=\sin 2x(\because 2\sin x\cos x=\sin 2x)$

Therefore,

$\frac{d}{dx}\left({\sin }^{2}x\right)=\sin 2x$