A well known identity of the Dirac delta function is that for any function $f(x)$: $$ \delta(x) f(x) = \delta(x) f(0). $$ If we take the derivative of the right hand side we get: $$ \delta'(x) f(0). $$ But if we take the derivative of the left hand side we get $$ \delta'(x) f(x) + \delta(x) f'(x) = \delta'(x) f(0) + \delta(x) f'(0) $$ Which one is correct?
P.S. I know that this problem has something to do with the fact that the delta function is not really a function, but rather a generalized function. However, the delta function and its derivatives are useful in calculations (especially in physics), and I want to know the correct rules for using them.