Question

53-56 Find \(y'\) and \(y''\).

53. \(y = {\bf{cos}}\left( {{\bf{sin}}\,{\bf{3}}\theta } \right)\)

Short answer

The value of \(y'\) is \( - 3\cos 3\theta \sin \left( {\sin 3\theta } \right)\).

The value of \(y''\) is \( - 9{\cos ^2}3\theta \cos \left( {\sin 3\theta } \right) + 9\sin 3\theta \sin \left( {\sin 3\theta } \right)\).

Step-by-step solution

Find the value of \(y'\)

Find the value of \(y'\) for the equation\(y = \cos \left( {\sin 3\theta } \right)\).

\(\begin{aligned}\frac{{{\rm{d}}y}}{{{\rm{d}}\theta }} &= \frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {\cos \left( {\sin 3\theta } \right)} \right)\\ &= - \sin \left( {\sin 3\theta } \right) \times \frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {\sin 3\theta } \right)\\ &= - \sin \left( {\sin 3\theta } \right) \times \left( {\cos 3\theta } \right) \times \frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {3\theta } \right)\\ &= - 3\cos 3\theta \sin \left( {\sin 3\theta } \right)\end{aligned}\)

So, the value of \(y'\) is \( - 3\cos 3\theta \sin \left( {\sin 3\theta } \right)\).

Find the value of \(y''\) 

Find the value of \(y''\) from the equation \(y' = - 3\cos 3\theta \sin \left( {\sin 3\theta } \right)\) using the product rule.

\(\begin{aligned}y'' &= \frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( { - 3\cos 3\theta \sin \left( {\sin 3\theta } \right)} \right)\\ &= - 3\cos 3\theta \frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {\sin \left( {\sin 3\theta } \right)} \right) - \sin \left( {\sin 3\theta } \right)\frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {3\cos 3\theta } \right)\\ &= - 3\cos 3\theta \left( {\cos \left( {\sin 3\theta } \right)} \right) \times \frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {\sin 3\theta } \right) - 3\sin \left( {\sin 3\theta } \right)\left( { - \sin 3\theta } \right)\frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {3\theta } \right)\\ &= - 3\cos 3\theta \left( {\cos \left( {\sin 3\theta } \right)} \right) \times \left( {\cos 3\theta } \right)\frac{{\rm{d}}}{{{\rm{d}}\theta }}\left( {3\theta } \right) + 9\sin 3\theta \sin \left( {\sin 3\theta } \right)\\ &= - 9{\cos ^2}3\theta \cos \left( {\sin 3\theta } \right) + 9\sin 3\theta \sin \left( {\sin 3\theta } \right)\end{aligned}\)

So, the value of \(y''\) is \( - 9{\cos ^2}3\theta \cos \left( {\sin 3\theta } \right) + 9\sin 3\theta \sin \left( {\sin 3\theta } \right)\).