What is the correct formula for [AB,C] in terms of A, B, and C?

[Caulfield]

Homework Statement

I am solving a problem and I arrived near the end, and can't figure out what to do here:

(1/(2m)) [P^2,X]+[P^2,X]

m - mass
P - Momentum operator
X - Position operator

Homework Equations

P = -iħ(∂/∂x)
[A,B]=AB-BA
[AB,C]=A[B,C]+B[A,C] where A, B and C are any operators

The Attempt at a Solution

I used [AB,C]=A[B,C]+B[A,C] to get 1/2m (P(PX-XP) + (PX-XP)P) and blocked

 

[strangerep]

I used [AB,C]=A[B,C]+B[A,C] to get 1/2m (P(PX-XP) + (PX-XP)P) and blocked

For an arbitrary wavefunction $\psi(x)$, what is $P\psi$ and $X\psi$? If you can figure those out, then you should be able to figure out $[X,P]\psi$ ...

 

[vanhees71]

The correct formula is
$$[AB,C]=A[B,C]+[A,C]B.$$