Functional derivative expansion (from Chaikin)

[redrzewski]

I'm self studying Chaikin's Principles of Condensed Matter Physics.
I'm trying to figure out how to go from (5.2.30) to (5.2.31).

Homework Statement

5.2.30 is the one-loop approx. to the free energy.

I'll denote G0^-1 from the book G
~ Integral(ln(G(phi(x)))

5.2.31 is (as far as I can tell) the 2nd term of the functional taylors expansion of this

Homework Equations

The Attempt at a Solution

So for the first functional derivative (i'll denote δ/δphi(y))

I get 1/G*δG/δ(phi(y))

Taking another δ/δphi(z) and I get:

1/G*δ^2G/δphi(y)δphi(z) for 1 term, which is one of the terms of 5.2.31, once you add in the integrals for the expansion.

However, I just can't get the G(x,x') term in 5.2.31.

For the other term of my expansion, I get instead:
-1/G^2*(δG/phi(y))*(δG/δphi(z))

 

[NateD]

Can anyone help me figure out how to get the G(x,x') term from 5.2.31? I'm really confused and any help would be greatly appreciated.Thank you!