Distributional derivative of $\left|\cos|\pi x|\right|$

Question

I am new to distributions and I have a problem finding the distributional derivative for the following function: $$\left|\cos|\pi x|\right|$$

I would be grateful if you could help to find the distributional derivative for this case. Please find the attached figure with this link. Problem

I have tried up to some extent and I have attached a figure of that one here too. However, I could not find a way to extend that solution. Please find that figure with this link.attempt_one

Answer 1

If $f$ is continuous and piecewise $C^1$ then its distributional derivative is just $g(a)=f'(a)$ at the $a$ where $f$ is differentiable.

Answer 2

Since $f(x)=|\cos|\pi x||=|\cos \pi x|$ is continuous and piecewise $C^1$ the distributional derivative is just the ordinary pointwise derivative where it's defined: $$f'(x) = \pi (-\sin \pi x) \operatorname{sign}(\cos\pi x).$$

The derivative is only piecewise continuous, piecewise $C^1$, and has steps from $-\pi$ to $+\pi$ at $x=\pm\frac12,\pm\frac32,\pm\frac52,\ldots$ It's derivative is therefore the pointwise derivative plus $-2\pi\delta(x-x_i)$ where $x_i=\pm\frac12,\pm\frac32,\pm\frac52,\ldots$: $$f''(x) = -\pi^2|\cos\pi x| + \sum_{k\in\mathbb{Z}}\delta(x-\frac{2k+1}{2}).$$