Parity operator commutes with second derivative?

[aaaa202]

How do I prove that the parity operator Af(x) = f(-x) commutes with the second derivative operator. I am tempted to write:

A∂^2f(x)/∂x^2 = ∂^2f(-x)/∂(-x)^2 = ∂^2f(-x)/∂x^2 = ∂^2Af(x)/∂x^2

But that looks to be abuse of notation..

 

[mfb]

You can do both differentiations separately, then it does not look so bad any more.