Dana G. answered • 10/08/19

Cornell Graduate for Mathematics and Science Tutoring

Finding this derivative requires the use of the chain rule. Notice you have two functions; first, the 'taking of the fourth power' on the inside and then the cosine function as the 'outside' function.

The derivative of cos(x) is -sin(x). But when using the chain rule, we must leave the inside stuff (x^{4}) inside the -sin( ... ) and then multiply by the derivative of the 'inside stuff', which is 4x^{3}. So we have:

-sin(x^{4})*4x^{3}

Which is the same as:

-4x^{3}sin(x^{4})

So f'(x)= -4x^{3}sin(x^{4})