Question

Differentiate the following questions w.r.t. x.

$\sqrt{e^{\sqrt{x}}}$

Answer 1

Let y =$\sqrt{e^{\sqrt{x}}}$

Fifferentiate both sides w.r.t. x,

$\Rightarrow \frac{dy}{dx}=\frac{1}{2}(e^{\sqrt{x}}{)}^{\frac{1}{2}-1}\frac{d}{dx}{e}^{\sqrt{x}}\Rightarrow \frac{dy}{dx}=\frac{1}{2}(e^{\sqrt{x}}{)}^{\frac{1}{2}}.e^{\sqrt{x}}.\frac{d}{dx}\left(\sqrt{x}\right)\Rightarrow \frac{dy}{dx}=\frac{1}{2}\frac{e^{\sqrt{x}}}{\sqrt{e^{\sqrt{x}}}}\times \frac{1}{2\sqrt{x}}=\frac{e^{\sqrt{x}}}{4\sqrt{x}\sqrt{e^{\sqrt{x}}}}=\frac{e^{\sqrt{x}}}{4\sqrt{x{e}^{\sqrt{x}}}}$