Derivative of $f(x) := \min(H(a+x),H(b-x))$ in the sense of distributions (Schwarz)

Question

Let $a$ and $b$ be real numbers and consider the function $f:\mathbb R \to \mathbb R$ defined by $$f(x) := \min(H(a+x),H(b-x)),$$ where $H:\mathbb R \to \mathbb R$ is the Heaviside function defined by $$ H(u) = \begin{cases}0,&\mbox{ if }u \le 0,\\ 1,&\mbox{ if }u>0. \end{cases} $$

Question. In the sense of distributions, what is the derivative of $f$ ?

I know that $H$ is locally-integrable and its derivative is the well-known Dirac distribution $\delta$. However, I'm very new to the theory of distributions, and I don't quite know how to go about the above problem.

Answer 1

$$ f^{\prime}(x)~=~H(a+b) [\delta(x+a)-\delta(x-b)] .$$