How does one prove that the Dirac Delta function satisfy this property ?
$$f(x)\delta'(x-y) =f(y)\delta'(x-y) - f(y)'\delta(x-y)$$
This is stated after this property
$$ f(x)\delta(x-y) =f(y)\delta(x-y)$$
which has been explained in this forum before in for example Why does the Dirac delta function satisfy $f(x)\delta(x-a) = f(a)\delta(x-a)$?
I have tried to use integration to prove it but all I got is $-f'(x)$ as final result, reference was https://www.reed.edu/physics/faculty/wheeler/documents/Miscellaneous%20Math/Delta%20Functions/Simplified%20Dirac%20Delta.pdf.
It seems closer to what one obtains with the for the Dirac delta derivative identity as in here Dirac delta derivative identity