Answer
\[\frac{{dy}}{{dx}} = {e^{ - x}}\cos x - {e^{ - x}}\sin x\]
Work Step by Step
\[\begin{gathered}
y = {e^{ - x}}\,\sin x \hfill \\
\hfill \\
use\,\,the\,\,product\,\,rule \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = {e^{ - x}}\,{\left( {\sin x} \right)^,} + \,\sin x{\left( {{e^{ - x}}} \right)^,} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = {e^{ - x}}\,\left( {\cos x} \right) + \sin x\,\left( { - {e^{ - x}}} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = {e^{ - x}}\cos x - {e^{ - x}}\sin x \hfill \\
\hfill \\
\end{gathered} \]
