Dirac delta derivative identity

Question

I have found the following identity online $f(t)\delta'(t) = f(0)\delta'(t) − f'(0) δ(t)$ however I am quite stuck on to how I would start this to prove this identity, I have an idea that I should use the distribution product rule however I can't seem to find it. Any help on how I can start this will be greatly appreciated hopefully I can continue and prove this identify.

Answer 1

By the product rule: $$ (f\delta)' = f'\delta + f\delta' $$

But $f\delta = f(0)\delta$ and $f'\delta = f'(0)\delta$ so $$ (f(0)\delta)' = f'(0)\delta + f\delta' $$ i.e. $$ f\delta' = (f(0)\delta)' - f'(0)\delta = f(0)\delta' - f'(0)\delta. $$