I have two related questions on the representation of the momentum operator in the position basis.

The action of the momentum operator on a wave function is to derive it:

$$\hat{p} \psi(x)=-i\hbar\frac{\partial\psi(x)}{\partial x}$$

(1) Is it ok to conclude from this that:

$$\langle x | \hat{p} | x' \rangle = -i \hbar \frac{\partial \delta(x-x')}{\partial x}?$$

And what does this expression mean?

(2) Using the equations:

$$ \frac{\langle x | \hat{x}\hat{p} | x' \rangle}{x} = \frac{\langle x | \hat{p}\hat{x} | x' \rangle}{x'} = \langle x | \hat{p} | x' \rangle $$

and

$$\langle x | [\hat{x},\hat{p}]|x'\rangle=i\hbar \delta(x-x')$$

one can deduce that

$$\langle x | \hat{p} | x' \rangle = i \hbar \frac{\delta(x-x')}{x-x'}$$

Is this equation ok? Does it follow that

$$\frac{\partial \delta(x-x')}{\partial x} = - \frac{\delta(x-x')}{x-x'}?$$