Chain rule on a dirac delta function

Question

I have the following expression that I would like to evaluate $$ \frac{\partial}{\partial\psi}\int_{y}\delta\left(f(x,y,\psi)\right)g(y)\,dy $$

My question is whether the chain rule behaves well when differentiating distributions rather than functions (I should note that my understanding of the dirac delta as a distribution, and how to differentiate it at all, remains extrememly shallow - I am actively following on links here, here and here to try and understand better). Below I've written what I naively might think to do $$ \int_{y}\frac{\partial}{\partial f}\delta\left(f(x,y,\psi)\right)\frac{\partial f}{\partial \psi}g(y)\,dy $$

I should note that the derivative $\frac{\partial f}{\partial \psi}$ is easy to get hold of. I should also note (though this may not be important) that $y$ is periodic and the integral over the domain of $y$ is actually only between $(-\pi, \pi]$.

Answer 1

I can see no problem with this as long as $f \in C^\infty.$ The Dirac $\delta$ can be approximated with ordinary functions for which the differentiation is valid, and then we just take limits.

Answer 2

Maybe you can use that $\delta(f(x))=\sum_{i=0}^n \frac{\delta(x-x_{i})}{|f'(x_{i}|}$ where $x_{i}$ is that $f(x_{i})=0$ and $f´(x_{i})\neq 0$ for more variables, don´t know the expression for that generalization of the property but maybe you can look it out