Question

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

Answer 1

`-2sin(x^2)-4x^2cos(x^2)`

Explanation:

This will require the chain rule. Recall that `d/dx[cos(u)]=-u'sin(u)`.

`y'=-d/dx[x^2]sin(x^2)`
`y'=-2xsin(x^2)`

To find the second derivative, we must use the product rule.

`y''=sin(x^2)d/dx[-2x]+(-2x)d/dx[sin(x^2)]`
`y''=-2sin(x^2)-2xcos(x^2)*d/dx[x^2]`
`y''=-2sin(x^2)-2xcos(x^2)*2x`
`y''=-2sin(x^2)-4x^2cos(x^2)`