Question

Differentiate the following w.r.t. $x$.
${\sin }^2\sqrt{x}$.

• A. $\frac{1}{2\sqrt{x}}\sin \left(3\sqrt{x}\right)$.
• B. $\frac{1}{\sqrt{x}}\sin \left(2\sqrt{x}\right)$.
  
• C. $\frac{1}{2\sqrt{x}}\sin \left(2\sqrt{x}\right)$.
• D. $\frac{1}{2\sqrt{x}}\sin \left(4\sqrt{x}\right)$.

Answer 1

The correct option is C $\frac{1}{2\sqrt{x}}\sin \left(2\sqrt{x}\right)$.

$\frac{d}{dx}{\sin }^2\sqrt{x}$

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

$=2\sin \sqrt{x}.\cos \sqrt{x}.\frac{1}{2\sqrt{x}}$

$=\sin \left(2\sqrt{x}\right).\frac{1}{2\sqrt{x}}[\because \sin 2\theta =2\sin \theta \cos \theta ]$