#### Answer

\[ = {e^{5x}}\csc x\,\left( {5 - \cot x} \right)\]

#### Work Step by Step

\[\begin{gathered}
y = {e^{5x}}\csc x \hfill \\
\hfill \\
Using\,\,product\,\,rule \hfill \\
\hfill \\
y' = \,{\left( {{e^{5x}}} \right)^\prime } \cdot \csc x + {e^{5x}} \cdot \,{\left( {\csc x} \right)^\prime } \hfill \\
\hfill \\
then \hfill \\
\hfill \\
= 5{e^{5x}}\csc x + {e^{5x}} \cdot \,\left( { - \csc x\,\cot x} \right) \hfill \\
\hfill \\
multiply \hfill \\
\hfill \\
= 5{e^{5x}}\csc x - {e^{5x}}\csc x\cot x \hfill \\
\hfill \\
factor \hfill \\
\hfill \\
= {e^{5x}}\csc x\,\left( {5 - \cot x} \right) \hfill \\
\end{gathered} \]