Answer
\[y' = {e^x}\cos \,\left( {\sin \,\left( {{e^x}} \right)} \right) \cdot \cos \,\left( {{e^x}} \right)\]
Work Step by Step
\[\begin{gathered}
y = \sin \,\left( {\sin \,\left( {{e^x}} \right)} \right) \hfill \\
\hfill \\
Chain\,\,rule \hfill \\
\hfill \\
f\,{\left( {g\,\left( t \right)} \right)^,} = {f^,}\,\left( {g\,\left( t \right)} \right) \cdot {g^,}\,\left( t \right) \hfill \\
\hfill \\
then \hfill \\
\hfill \\
{y^,} = {\sin ^,}\,\left( {\sin \,\left( {{e^x}} \right)} \right) \cdot \,{\left( {\sin \,\left( {{e^x}} \right)} \right)^,} \hfill \\
\hfill \\
y' = \cos \,\left( {\sin \,\left( {{e^x}} \right)} \right) \cdot \cos \,\left( {{e^x}} \right) \cdot \,{\left( {{e^x}} \right)^,} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
y' = {e^x}\cos \,\left( {\sin \,\left( {{e^x}} \right)} \right) \cdot \cos \,\left( {{e^x}} \right) \hfill \\
\end{gathered} \]
