Functional Derivatives: Overview & Tips

[The black vegetable]

TL;DR

Can someone please explain how you take a functional derivative of a generating function. I have included a basic example of how I understand it, if it's not the case can someone explain explicitly for a dummy how this is performed. My question is in picture form, as it's much easier for me to do it this way.
Many thanks

Hi

In the last sentence I mean you do include constant terms like I have done when taking the product above?

 

[A. Neumaier]

Replace $J$ by $J+\epsilon dJ$, expand, drop terms involving higher powers of $\epsilon$. The coefficient of $\epsilon$ is the integral of the functional derivative $dZ(J)/dJ(x)$ multiplied by $dJ(x)$.