In these notes, they use some reasoning to get 10.1, that is $\delta(ax)=\delta(x)/|a|$. I am able to follow that, but then they proceed from there to show (11), which I am having some trouble understanding. Could someone please clarify this?
Addendum: They then generalize it for a function with more than two roots in (12), but that doesn't even seem to work for the $\Theta(x^2-a^2)$ example they have, as using the first line I get $0$ for $x<-a$ and $x>a$, and $1$ for $-a<x<a$.
I am aware there are other demonstrations of identity (11), but I am specifically interested in this one.
They have written $\theta(x^2-a^2) = 1 - \theta(x+a) + \theta(x-a)$ (assuming $a>0$) and then taken the derivative of both sides using the chain rule on the left hand side: $$2x\,\delta(x^2-a^2) = -\delta(x+a) + \delta(x-a).$$ Then they have divided with $2x$: $$\delta(x^2-a^2) = -\frac{1}{2x}\delta(x+a) + \frac{1}{2x}\delta(x-a).$$ At this step they have used that $f(x)\,\delta(x-a) = f(a)\,\delta(x-a)$ to get $$\delta(x^2-a^2) = -\frac{1}{2(-a)}\delta(x+a) + \frac{1}{2a}\delta(x-a) = \frac{1}{2a}\delta(x+a) + \frac{1}{2a}\delta(x-a).$$