Question

If $y=e^{\sin \sqrt{x}},$ then $\frac{dy}{dx}=$

• A. $\frac{e^{\cos \sqrt{x}}\sin \sqrt{x}}{2\sqrt{x}}$
• B. $\frac{e^{\sin \sqrt{x}}}{2\sqrt{x}}$
• C. $\frac{e^{\sin \sqrt{x}}\cos \sqrt{x}}{2\sqrt{x}}$
• D. $\frac{e^{\sin \sqrt{x}}\cos \sqrt{x}}{\sqrt{x}}$

Answer 1

The correct option is C $\frac{e^{\sin \sqrt{x}}\cos \sqrt{x}}{2\sqrt{x}}$

We have,
$y=e^{\sin \sqrt{x}}$
Differentiate it with respect to $x,$ $\frac{dy}{dx}=\frac{d}{dx}{e}^{\sin \sqrt{x}}$
$=e^{\sin \sqrt{x}}\times \frac{d}{dx}\left(\sin \sqrt{x}\right)\text{[Using chain rule ]}$
$=e^{\sin \sqrt{x}}\times \cos \sqrt{x}\times \frac{d}{dx}\sqrt{x}\text{[Using chain rule ]}$
$=e^{\sin \sqrt{x}}\times \cos \sqrt{x}\times \frac{1}{2\sqrt{x}}$
$=\frac{e^{\sin \sqrt{x}}\cos \sqrt{x}}{2\sqrt{x}}$