quantum mechanics - Commutator $[hat{p},F(hat{x})]$ of Momentum $hat{p}$ with a Position dependent function $F(hat{x})$?

I heard from my GSI that the commutator of momentum with a position dependent quantity is always $-i\hbar$ times the derivative of the position dependent quantity. Can someone point me towards a derivation, or provide one here?

Answer

You start from this

$[p,F(x)]\psi=(pF(x)-F(x)p)\psi$

knowing that $p=-i\hbar\frac{\partial}{\partial x}$ you'll get

$[p,F(x)]\psi=-i\hbar\frac{\partial}{\partial x}(F(x)\psi)+i\hbar F(x)\frac{\partial }{\partial x}\psi=-i\hbar\psi\frac{\partial}{\partial x}F(x)-i\hbar F(x)\frac{\partial}{\partial x}\psi+i\hbar F(x)\frac{\partial }{\partial x}\psi$

from where you find that $[p,F(x)]=-i\hbar\frac{\partial}{\partial x}F(x)$