dy/dx = -2xsinx^2dydx=−2xsinx2
Process:
This problem will require use of the chain rule.
If y = cosx^2y=cosx2, then, by the chain rule, the derivative will be equal to the derivative of cosx^2cosx2 with respect to x^2x2, multiplied by the derivative of x^2x2 with respect to xx.
We know the basic identity d/(dx)[cos x] = -sin xddx[cosx]=−sinx. And, the power rule gives us d/(dx) [x^2] = 2xddx[x2]=2x.
(if those identities look unfamiliar to you, some excellent videos can be located here and here, which explain the identity for cos xcosx and the power rule, respectively)
So, the derivative of cosx^2cosx2 will therefore be:
d/(dx) [cos x^2] = -sinx^2 * d/dx[x^2]ddx[cosx2]=−sinx2⋅ddx[x2]
Which further simplifies to:
d/dx [cos x^2] = -2xsin x^2ddx[cosx2]=−2xsinx2
