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