#### Answer

\[\frac{{dy}}{{dx}} = - 4{e^x}\,{\left( {1 - {e^x}} \right)^3}\]

#### Work Step by Step

\[\begin{gathered}
y = \,{\left( {1 - {e^x}} \right)^4} \hfill \\
\hfill \\
Use\,\,the\,\,version\,\,1\,\,of\,\,the\,\,chain\,\,rule \hfill \\
\hfill \\
{\text{ }}\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}} \hfill \\
\hfill \\
set\,\,u = 1 - {e^x} \hfill \\
\hfill \\
\frac{{du}}{{dx}} = - {e^x} \hfill \\
\hfill \\
therefore \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = 4\,{\left( {1 - {e^x}} \right)^3}\,\left( { - {e^x}} \right) \hfill \\
\hfill \\
multiply \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = - 4{e^x}\,{\left( {1 - {e^x}} \right)^3} \hfill \\
\end{gathered} \]