What is the derivative of f(x) = cos (x^4)?

Answer 1

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:

-sin(x⁴)*4x³

Which is the same as:

-4x³sin(x⁴)

So f'(x)= -4x³sin(x⁴)