A Functional Derivatives in Q.F.T.

[QFT1995]

I'm can't seem to figure out how to functionally differentiate a functional such as Z(J)= e^{\frac{i}{2} \int \mathrm{d}^4y \int \mathrm{d}^4x J(y) G_F (x-y) J(x)}
with respect to J(x). I know the answer is
\frac{\delta Z(J)}{\delta J(x)}= -i \int \mathrm{d}^4y J(y) G(x-y)
but I'm struggling to calculate it.

 

[A. Neumaier]

The general recipe for calculating functional derivatives is:

Change each occurrence of $J(x)$ for some $x$ to $J(x)+sf(x)$ with a test function $f(x)$, then differentiate with respect to $s$, set $s=0$ in the result, and take the limit where $f(x)$ tends to the Dirac delta function.