In Srednicki pg.136, he derives the Schwinger-dyson equation from: $$ 0=\delta Z(J)=i\int D\phi \, e^{i[S+\int d^4y\, J_b\phi_b]}\int d^4x(\frac{\delta S}{\delta \phi_a}+J_a)\delta\phi_a.\tag{22.21} $$
Then we apply $n$ functional derivatives wrt. $J$ on it, and we can get: $$ \begin{align}0=&\int D\phi e^{iS}\int d^4x[\,i\frac{\delta S}{\delta \phi_a}\phi_{a_1}...\phi_{a_n}\cr&+\sum_{j=1}^{n} \phi_{a_1}(x_1)...\delta_{aa_j}\delta^4(x-x_j)...\phi_{a_n}(x_j)]\delta\phi_a\tag{22.22} \end{align}$$ after setting $J=0$. But I am stuck when I try to go through this procedure. For example, I simply do one functional derivatives: $$ \frac{\delta}{\delta J_{a_1}(x_1)}Z(J)=\int D\phi\,e^{i[S+\int d^4x\,J_a\phi_a]}\frac{\delta}{\delta J_{a_1}(x_1)}i(S+\int d^4x J_a\phi_a). $$ And I am confused by the $S[\phi]$ term, how can it become of the form $\frac{\delta S}{\delta\phi_a}\phi_{a}$? I think the $S[\phi]$ term must be disappear under the derivatives with respect to $J$. Where does the $\phi_a$ term come from?
The right answer: $$ \frac{\delta}{\delta J_{a_1}(x_1)} \delta Z(J) =\int D\phi...[\phi_{a_1}(x_1)\int d^4x(\frac{\delta S}{\delta\phi}+J)+\int d^4x\delta_{aa_1}\delta^4(x-x_1)]\delta\phi. $$