OK I found the answer. To get rid of the derivatives of Dirac deltas, I used the identity [itex]f(x) \delta ' (xy) = f(y) \delta ' (xy)  f'(x) \delta (xy)[/itex] which follows from the convolution between delta and [itex](f \times \varphi)'[/itex] ([itex]\varphi[/itex] is a test function).
This also produces the remaining wanted term.
