Question

7-52 Find the derivative of the function.

52. \(y = {\bf{si}}{{\bf{n}}^{\bf{3}}}\left( {{\bf{cos}}\left( {{x^{\bf{2}}}} \right)} \right)\)

Short answer

The derivative of the given equation is \( - 6x\sin \left( {{x^2}} \right){\sin ^2}\left( {\cos \left( {{x^2}} \right)} \right)\cos \left( {\cos \left( {{x^2}} \right)} \right)\).

Step-by-step solution

Recall the chain rule 

According to the chain rule, the derivative of a function \(F\left( x \right) = f\left( {g\left( x \right)} \right)\) can be obtained as:

\(\begin{aligned}F'\left( x \right) &= f'\left( {g\left( x \right)} \right)\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {g\left( x \right)} \right)\\ &= f'\left( {g\left( x \right)} \right)g'\left( x \right)\end{aligned}\)

Find the derivative of the given function 

The derivative of the equation \(y = {\sin ^3}\left( {\cos \left( {{x^2}} \right)} \right)\).

\(\begin{aligned}\frac{{{\rm{d}}y}}{{{\rm{d}}x}} &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{{\sin }^3}\left( {\cos \left( {{x^2}} \right)} \right)} \right)\\ &= 3{\sin ^2}\left( {\cos \left( {{x^2}} \right)} \right) \cdot \cos \left( {\cos \left( {{x^2}} \right)} \right) \times \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\cos \left( {{x^2}} \right)} \right)\\ &= 3{\sin ^2}\left( {\cos \left( {{x^2}} \right)} \right) \cdot \cos \left( {\cos \left( {{x^2}} \right)} \right) \times \left( { - \sin \left( {{x^2}} \right)} \right) \times \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{x^2}} \right)\\ &= - 3\sin \left( {{x^2}} \right){\sin ^2}\left( {\cos \left( {{x^2}} \right)} \right)\cos \left( {\cos \left( {{x^2}} \right)} \right)\left( {2x} \right)\\ &= - 6x\sin \left( {{x^2}} \right){\sin ^2}\left( {\cos \left( {{x^2}} \right)} \right)\cos \left( {\cos \left( {{x^2}} \right)} \right)\end{aligned}\)

Thus, the derivative of the given equation is \( - 6x\sin \left( {{x^2}} \right){\sin ^2}\left( {\cos \left( {{x^2}} \right)} \right)\cos \left( {\cos \left( {{x^2}} \right)} \right)\).