understanding an argument involving derivative and integral of a dirac delta generalised function

Question

An argument about moving the one-dim wave equation in vector space to position space involves the below equality.

Note: the primes on the variable x do not denote derivatives of any sort. They are merely placeholders for distinguishing different positions.

$\frac{\partial }{\partial x}\left ( \int_{a}^{b}\int_{a}^{b} \delta \left ( x-x' \right )\left ( \frac{\partial }{\partial x}\delta \left ( x'-x'' \right ) \right )f\left ( x'',t \right )dx'dx'' \right )$

leads to

$\frac{\partial }{\partial x}\int_{a}^{b}\left ( \frac{\partial }{\partial x} \delta \left ( x-x'' \right )\right )f\left ( x'',t \right )dx''$

I would like to understand the above argument. Any illumination would be helpful.

Answer 1

We have for any $g$ $$\int_a^b \delta(x-x') g(x'-x'') dx' = g(x-x'')$$ Now take $g(x'-x'') = \frac{\partial}{\partial x} \delta(x'-x'')$ and you have your result.