Question

$\frac{d}{dx}\left[\sqrt{\sin \sqrt{x}}\right]=$

• A. $\frac{\cos \sqrt{x}}{4\sqrt{x}\sqrt{\sin \sqrt{x}}}$
• B. $\frac{\sin \sqrt{x}}{4\sqrt{x}\sqrt{\sin \sqrt{x}}}$
• C. $\sin \sqrt{x}$
• D. $\cos \sqrt{x}$

Answer 1

The correct option is A $\frac{\cos \sqrt{x}}{4\sqrt{x}\sqrt{\sin \sqrt{x}}}$

Using chain rule:
$\frac{d}{dx}\left[\sqrt{\sin \sqrt{x}}\right]=\frac{1}{2}[\sin \sqrt{x}{]}^{\frac{1}{2}-1}\times \frac{d}{dx}(\sin \sqrt{x})$
$=\frac{1}{2\sqrt{\sin \sqrt{x}}}.\left(\cos \sqrt{x}\right)\times \frac{d}{dx}\sqrt{x}=\frac{\cos \sqrt{x}}{2\sqrt{\sin \sqrt{x}}}\times \frac{1}{2\sqrt{x}}=\frac{\cos \sqrt{x}}{4\sqrt{x}\sqrt{\sin \sqrt{x}}}$