#### Answer

\[\frac{{dy}}{{dx}} = \frac{{{e^{\sqrt x }}}}{{2\sqrt x }}\]

#### Work Step by Step

\[\begin{gathered}
y = {e^{\sqrt x }} \hfill \\
\hfill \\
use\,\,\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}}\,\,\, \hfill \\
\hfill \\
set\,\,u = \sqrt x \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = {e^{\sqrt x }}\,{\left( {\sqrt x } \right)^,} \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = {e^{\sqrt x }}\,\left( {\frac{1}{{2\sqrt x }}} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \frac{{{e^{\sqrt x }}}}{{2\sqrt x }} \hfill \\
\end{gathered} \]