I have an integral
$$ \Psi(x,t) = \frac{1}{(2 \pi \hbar)^{3/2}} \int \Phi(p,t) \exp(-\frac{i}{\hbar}px) d^3p $$
and I want to substitute $p$ with $-i \hbar \nabla$. So how do I rewrite $d^3p$? In my understanding it's going to look like
$$ d^3 p = (-i \hbar)^3 (d \nabla)^3 $$ and then? How do I proceed? I want that the integral in the end looks something like $$ \int ... d^3x $$
I looks for me that you are doing a fourier transformation, but regarding your substitution it is not clear what you mean. First I thougt, by putting a derivation in the integral you could mean the use of differentiation rules for the fourier transformation (see here), but by your last line you maybe mean the inverse fourier transformation. This is what physicist often do, when they switch between position and momentum space, they simly fourier transform.
Hopefully one of the two things is anything which brings your closer to your aim. Maybe you could more clarify your question with all the ideas you got so far (also by the comments).