Behaviour of an Exponential Commutation

[d.hatch75]

Homework Statement

Forgive the awkward title, it was hard to think how to describe the problem in such a short space. I'm following the proof of Proposition 1.2 of Nakahara's "Geometry, Topology and Physics", and the following commutation relation has been established:

$\partial_x^n e^{ikx} = e^{ikx}(ik + \partial_x )^n$

The line immediately afterwards is:

$e^{-i\epsilon \{-\partial_x^2 / 2m + V(x)\}}e^{ikx} = e^{ikx}e^{-i\epsilon \{-(ik+\partial_x )^2 / 2m + V(x)\}}$

I am not seeing how this follows from the previous relation, since the partial derivative is in the exponential so surely it doesn't act on $e^{ikx}$ in the same way? It seems dodgy, but I can only assume I'm missing some important result concerning exponentials of commutation relationships that hasn't otherwise been specified anywhere in the book as far as I can tell.

Homework Equations

(See above)

The Attempt at a Solution

(See above)

 

[BruceW]

hint: try using the power series expansion of an exponential function.