Derivative in the sense of distributions

Question

I have a question regarding calculating the derivative in the distribution sense of the following function:

$$ f(x) = \frac{d^2 }{d x^2}|\cos|x|| $$

Maybe someone can point me in the right direction.

Thanks, Daniel.

Answer 1

You must guess which is the derivative. Since $|\cos x|$ is differentiable almost everywhere, the pointwise derivative should work. Then you must verify that the pointwise derivative satisfies the definition of weak derivatives.