Chapter 3 - Derivatives - 3.7 The Chain Rule - 3.7 Exercises - Page 192: 49

$y' = - 15{\sin ^4}\,\left( {\cos 3x} \right) \cdot \cos \,\left( {\cos 3x} \right) \cdot \sin 3x$

Work Step by Step

$\begin{gathered} y = {\sin ^5}\,\left( {\cos 3x} \right) \hfill \\ \hfill \\ Chain\,\,rule \hfill \\ \hfill \\ f\,{\left( {g\,\left( t \right)} \right)^,} = {f^,}\,\left( {g\,\left( t \right)} \right) \cdot {g^,}\,\left( t \right) \hfill \\ \hfill \\ {y^,} = 5{\sin ^4}\,\left( {\cos 3x} \right) \cdot \,{\left( {\sin \,\left( {\cos 3x} \right)} \right)^,} \hfill \\ \hfill \\ use\,\,\frac{d}{{dx}}\left[ {\sin u} \right] = u'\cos u \hfill \\ \hfill \\ y' = 5{\sin ^4}\,\left( {\cos 3x} \right) \cdot \cos \,\left( {\cos 3x} \right) \cdot \,{\left( {\cos 3x} \right)^,} \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ y' = 5{\sin ^4}\,\left( {\cos 3x} \right) \cdot \cos \,\left( {\cos 3x} \right) \cdot \,\,\left( { - \sin 3x} \right) \cdot \,{\left( {3x} \right)^,} \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ y' = 5{\sin ^4}\,\left( {\cos 3x} \right) \cdot \cos \,\left( {\cos 3x} \right) \cdot \,\sin 3x \cdot 3 \hfill \\ \hfill \\ y' = - 15{\sin ^4}\,\left( {\cos 3x} \right) \cdot \cos \,\left( {\cos 3x} \right) \cdot \sin 3x \hfill \\ \end{gathered}$

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