Let $\mathrm{u}=\mathrm{e}^x$ and $\mathrm{v}=\sqrt{x}$ Differentiating u and v w.r.t. $x$, we get
$\begin{aligned}
& \frac{\mathrm{du}}{\mathrm{dx}}=\mathrm{e}^x, \frac{\mathrm{dv}}{\mathrm{~d} x}=\frac{1}{2 \sqrt{x}} \\
\therefore \quad & \frac{\mathrm{du}}{\mathrm{dv}}=\frac{\frac{\mathrm{du}}{\mathrm{dx}}}{\frac{\mathrm{dv}}{\mathrm{dx}}}=\frac{\mathrm{e}^x}{\frac{1}{2 \sqrt{x}}}=2 \mathrm{e}^x \sqrt{x}
\end{aligned}$