How do you find the derivative of #y=cos(x^2)# ?

1 Answer
Aug 4, 2014

#dy/dx = -2xsinx^2#

Process:

This problem will require use of the chain rule.

If #y = cosx^2#, then, by the chain rule, the derivative will be equal to the derivative of #cosx^2# with respect to #x^2#, multiplied by the derivative of #x^2# with respect to #x#.

We know the basic identity #d/(dx)[cos x] = -sin x#. And, the power rule gives us #d/(dx) [x^2] = 2x#.

(if those identities look unfamiliar to you, some excellent videos can be located here and here, which explain the identity for #cos x# and the power rule, respectively)

So, the derivative of #cosx^2# will therefore be:

#d/(dx) [cos x^2] = -sinx^2 * d/dx[x^2]#

Which further simplifies to:

#d/dx [cos x^2] = -2xsin x^2#