Question

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

Answer 1

`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`