Ordinary function is a function in the standard sense, albeit not necessarily continuous. F.e. Heaviside function. As $x\ne 0$ we can differentiate it, producing $\overset{\circ}{f}{}'$, but we also can find it's derivative as a distribution, producing $f'(x)$. What is $\overset{\circ}{f}{}'$ here? And what is $f'$ here?

Let $$f(x)=\left\{\begin{aligned}&\cos(x) &&x>0,\\

&0 &&x\le 0\end{aligned}\right.$$

What is $\overset{\circ}{f}{}'$ here? And what is $f'$ here?