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 (x4) inside the -sin( ... ) and then multiply by the derivative of the 'inside stuff', which is 4x3. So we have:
Which is the same as:
So f'(x)= -4x3sin(x4)