Answer
\[\frac{{dy}}{{d\theta }} = - 4\sin \theta {\cos ^3}\theta + 4\cos \theta {\sin ^3}\theta \]
Work Step by Step
\[\begin{gathered}
y = {\cos ^4}\theta + {\sin ^4}\theta \hfill \\
\hfill \\
Use\,\,the\,\,version\,\,2\,\,of\,\,the\,\,chain\,\,rule \hfill \\
\hfill \\
\,\,y = {u^n} \to \frac{{dy}}{{dx}} = n{u^{n - 1}}{u^,} \hfill \\
\hfill \\
then \hfill \\
\hfill \\
\frac{{dy}}{{d\theta }} = 4{\cos ^3}\,\left( \theta \right)\,\left( { - \sin \theta } \right) + 4{\sin ^3}\,\left( \theta \right)\,\left( {\cos \theta } \right) \hfill \\
\hfill \\
simplify \hfill \\
\hfill \\
\frac{{dy}}{{d\theta }} = - 4\sin \theta {\cos ^3}\theta + 4\cos \theta {\sin ^3}\theta \hfill \\
\end{gathered} \]
