Use chain rule to compute the derivatives.

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}={\frac{{{d}}}{{{d}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}\times{\frac{{{d}}}{{{d}{\left({f{{\left({x}\right)}}}\right)}}}}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}\times{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{f{{\left({x}\right)}}}\)

\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{f}'{\left({x}\right)}\)

\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{g{{\left({x}\right)}}}\)

\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}{g{{\left({x}\right)}}}{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\)

\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}={\frac{{{d}}}{{{d}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}}}}{\left({e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\right)}\times{\frac{{{d}}}{{{d}{\left({f{{\left({x}\right)}}}\right)}}}}{\left({\sin{{\left({f{{\left({x}\right)}}}\right)}}}\right)}\times{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{f{{\left({x}\right)}}}\)

\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{f}'{\left({x}\right)}\)

\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}\times{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\times{g{{\left({x}\right)}}}\)

\(\displaystyle={e}^{{{\sin{{\left({f{{\left({x}\right)}}}\right)}}}}}{g{{\left({x}\right)}}}{\cos{{\left({f{{\left({x}\right)}}}\right)}}}\)