How do you use the chain rule to differentiate \[y={{\sin }^{2}}\left( \cos \left( 4x \right) \right)\]?

Hint: To differentiate, we have to use chain rule which states that the derivative of \[f\left( g\left( x \right) \right)\] is \[f'\left( g\left( x \right) \right).g'\left( x \right)\]. Also we use chain rules to differentiate composite functions. Composite function is created when \[1\] function is substituted in another example of composite function \[f\left( g\left( x \right) \right)\].

Complete Step by Step solution:
Here, we have \[y={{\sin }^{2}}\left( \cos \left( 4x \right) \right)\]
Let \[f\left( x \right)={{\sin }^{2}}\left( x \right)\] and \[g\left( x \right)=\cos \left( 4x \right)\]
Thus, we get \[y=f\left( g\left( x \right) \right)={{\sin }^{2}}\left( \cos \left( 4x \right) \right)\]
Now, we can observe that the above function is a composite function.
So, here we can apply the chain rule.
\[\therefore \] By chain rule we get,
\[\dfrac{df\left( g\left( x \right) \right)}{dx}=f'\left( g\left( x \right) \right).g'\left( x \right)\]
\[=2\sin \left( \cos \left( 4x \right) \right).cos\left( \cos \left( 4x \right) \right)\]
\[\Rightarrow 2\sin \left( \cos \left( 4x \right) \right).cos\left( \cos \left( 4x \right) \right)\] has a composite function as \[\cos \left( \cos \left( 4x \right) \right)\]
Thus, by applying chain rule again we get,
\[\dfrac{dy}{dx}=2\sin \left( \cos \left( 4x \right) \right).cos\left( \cos \left( 4x \right) \right).\dfrac{d}{dx}\cos \left( 4x \right)\]
Here, again we can observe that, \[\cos \left( 4x \right)\] is a composite function. So, we will again apply the chain rule, we get,
\[\dfrac{dy}{dx}=2\sin \left( \cos \left( 4x \right) \right).cos\left( \cos \left( 4x \right) \right).\left( -\sin \left( 4x \right) \right).4\]

\[=-8\sin \left( 4x \right)\sin \left( \cos \left( 4x \right) \right)\cos \left( \cos \left( 4x \right) \right)\]

Note:
In differentiating such problems after solving the composite function we can again get the composite function thus, we have to keep simplifying the composite functions by chain rule. In case if the composite function continues then here we use the concept of ‘partial differentiation’. By this it can be simplified.