Functional derivative of generating function QFT I am reading Srednicki book for QFT. In page 69 they realize a functional derivative of the following function: $$ Z[J]= \exp\left(\frac{i}{2}\int d^4xd^4x'J(x)\Delta(x-x')J(x') \right) $$ with $$\Delta(x-x')=\lim_{\epsilon\rightarrow 0}\int \frac{d^4k}{(2\pi)^4}\frac{\exp\\{ik(x-x')\\}}{k^2+m^2-i\epsilon}$$ the usual Feynman propagator. At some point they say that $$ \frac{\delta}{\delta(J(x_1))}Z[J]=iZ[J]\int d^4x'\Delta(x_1-x')J(x') $$ I've tried to derive that result but I am not use to functionals having two integrates and I am not very used to functional derivatives. Any help for the derivation of this result is appreciated.

**HINT**

In these situations, it's usually helpful to try to translate it into a discrete context and see if you can make sense of it there. For instance, let $$ Z(J) = \exp\left(\frac{i}{2} \sum_{i,j}J_i \Delta_{ij}J_j\right).$$

Can you see that $$\frac{\partial}{\partial J_k}Z(J) = i Z(J) \sum_j\Delta_{kj}J_j $$ using the fact that $\Delta_{ij}$ is symmetric?