#### Answer

\[ = \frac{{{e^x}}}{x} + {e^x}\ln x\]

#### Work Step by Step

\[\begin{gathered}
\frac{d}{{dx}}\,\left( {{e^x}\ln x} \right) \hfill \\
\hfill \\
Use\,\,product\,\,rule \hfill \\
\hfill \\
= {e^x}\frac{d}{{dx}}\,\,\left[ {\ln x} \right] + \ln x\frac{d}{{dx}}\,\left( {{e^x}} \right) \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
= {e^x}\,\left( {\frac{1}{x}} \right) + \ln x\,\left( {{e^x}} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
= \frac{{{e^x}}}{x} + {e^x}\ln x \hfill \\
\end{gathered} \]