Answer
\[{f^,}\,\left( x \right) = \frac{{\cos x\,\,{e^{\sin x}}}}{{\sqrt {1 - {e^{2\sin x}}} }}\]
Work Step by Step
\[\begin{gathered}
\hfill \\
f\,\left( x \right) = {\sin ^{ - 1}}\,\left( {{e^{\sin x}}} \right) \hfill \\
\hfill \\
Use\,\,the\,\,formula\,\,\frac{d}{{dx}}\,\,\left[ {{{\sin }^{ - 1}}u} \right] = \frac{{{u^,}}}{{\sqrt {1 - {u^2}} }} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
{f^,}\,\left( x \right) = \frac{{\cos x\,\,\,{e^{\sin x}}}}{{\sqrt {1 - \,\,{{\left[ {{e^{\sin x}}} \right]}^2}} }} \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
{f^,}\,\left( x \right) = \frac{{\cos x\,\,{e^{\sin x}}}}{{\sqrt {1 - {e^{2\sin x}}} }} \hfill \\
\end{gathered} \]
