#### Answer

\[\frac{{dy}}{{dx}} = - 2x{e^{ - {x^2}}}\]

#### Work Step by Step

\[\begin{gathered}
y = {e^{ - {x^2}}} \hfill \\
\hfill \\
Use\,\,the\,\,version\,\,1\,\,of\,\,the\,\,chain\,\,rule \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}} \hfill \\
\hfill \\
set\,\,u = - {x^2} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = {e^{ - {x^2}}}\frac{d}{{dx}}\,\left( { - {x^2}} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = - 2x{e^{ - {x^2}}} \hfill \\
\hfill \\
\end{gathered} \]