Scaling of the delta function derivative

Question

I'm stuck figuring out a simple scaling property for the derivative of the delta function.

What relation am i missing that results in

$$ \delta'(ax) = \frac{1}{a^2}\delta'(x) $$

Instead of just $\frac{d}{dx}(\delta(ax)) = \frac{1}{|a|}\delta'(x)$?

Answer 1

Note that $\delta'(ax)$ and $\left(\delta(ax)\right)'$ are two different expressions (e.g. if $f(x) = x^2$, then $f'(2x) = 4x$ but $(f(2x))' = 8x$).

Informally,

$$ \int \delta'(ax)\phi(x)\,dx = \frac{1}{a}\int \delta'(y)\phi\left(\frac ya\right)\, dy = -\frac1a\int \delta(y) \left(\phi\left(\frac ya\right)\right)'\,dy \\ = -\frac{1}{a^2}\int \delta(y) \phi'\left(\frac ya\right)\,dy = -\frac{1}{a^2}\phi'(0). $$