#### Answer

\[\frac{{dy}}{{dx}} = - 2\cos x\sin x\]

#### Work Step by Step

\[\begin{gathered}
y = {\cos ^2}x \hfill \\
\hfill \\
or \hfill \\
y = \left( {\cos x} \right)\left( {\cos x} \right) \hfill \\
\hfill \\
\,\,use\,\,the\,\,product\,\,rule. \hfill \\
\hfill \\
\frac{{dx}}{{dy}} = 2\left( {\cos x\,} \right)\,{\left( {\cos x} \right)^\prime } + \left( {\cos x\,} \right)\,{\left( {\cos x} \right)^\prime } \hfill \\
\hfill \\
\frac{{dx}}{{dy}} = 2\,\left( {\cos x\,} \right)\,{\left( {\cos x} \right)^\prime } \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{dx}}{{dy}} = 2\,\cos x\,\left( { - \sin x} \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{dx}} = - 2\cos x\sin x \hfill \\
\hfill \\
\end{gathered} \]