Question

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

• A. $sin2x$
• B. $cos2x$
• C. $cos4x$
• D. $sin4x$

Answer 1

Answer: (A)

$y={\sin }^{2}x$
$\frac{dy}{dx}=\underset{\delta x\to 0}{lim}\frac{{\sin }^2(x+\delta x)-{\sin }^{2}x}{\delta x}$
$=\underset{\delta x\to 0}{lim}\frac{{\sin }^2(x+\delta x+x).\sin (x+\delta x-x)}{\delta x}$
$=\underset{\delta x\to 0}{lim}\sin (2x+\delta x)\frac{\sin \delta x}{\delta x}=\sin 2x.$
We have used
$[{\sin }^{2}A-{\sin }^{2}B=\sin (A+B)\sin (A-B)]$