I have very little background with functional derivatives and I would like to clarify some issues.
I am trying to compute the second functional derivative of the Klein Gordon action expressed in real components
I know how to compute the first functional derivative
from now on I have doubts. I wanna compute
If there were no derivatives this would be trivial but I don't know how to deal with $\partial^2$. I have thought in introducing a delta to write this as an integral and use the formula I have used to compute the first functional derivative, but I am concerned about taking derivatives of something having a Dirac delta.
So, my question is, how am I supposed to compute this functional derivative?