How to prove $\delta [g(x)]= \sum_i \frac{\delta (x-x_i)}{|g'(x_i)|} $ [duplicate]

I've seen this identity in my electrodynamics book: $$\delta (f(x))=\sum_i{ \frac{1}{|{df\over dx}(x_i)|}\delta (x-x_i)}$$

Where $x_i$ shows the $i$th zero of $f(x)$. How can I prove it? I've tried the integral definition of delta function, but doesn't work.

Here's an informal idea:

Start with the integral

$$ \int \delta(f(x)) g(x)\,dx $$

and for every $x_i$, take disjoint neighborhoods $U_i$ where $f$ is a diffeomorphism (i.e. $f' \neq 0$). So,

$$ \int \delta(f(x)) g(x)\,dx = \sum_i \int_{U_i} \delta(f(x)) g(x)\,dx $$

use change of variables in each neighborhood: $u_i = f(x)$ so $$ \int \delta(f(x)) g(x)\,dx = \sum_i \int_{f(U_i)} \delta(u_i) \frac{g(f^{-1}(u_i))}{|f'(f^{-1}(u_i))|}\,du $$

then $u_i = 0$ exactly when $x = x_i$, so we have

$$ \int \delta(f(x)) g(x)\,dx = \sum_i \frac{g(x_i)}{|f'(x_i)|} $$