I know that if we have some function f(y) that is smooth and has compact support, we get the following by using integration by parts.
$\int f(y) \delta '(y-x) dy = -\int f'(y) \delta (y-x) dy =-f'(x)$
And more generally we have the following statement.
$\int f(y) \delta ^{(n)}(y-x) dy = (-1)^n f^{(n)}(x)$
My problem is currently what do I do if $f(y)$ is not smooth. I am dealing with a case where $f'(y)$ isn't defined at the point $x$. For example, if I have $f(y) = 1/y$ and I'm considering the following integral is there anything that I can say about the value of this integral?
$\int \frac{1}{y} \delta ^{(n)}(y) dy$
In this case $f'(y) = -\frac{1}{y^2}$ which is undefined at $y=0$.
I've tried plugging the above integral into Mathematica for $n=1,2,3,4$ and I have gotten $0$ every time. I'm just not sure why that answer would make sense. Here is an example of the code I used
Integrate[Derivative[4][DiracDelta][s]*(1/s), {s, 0, Infinity}]