Deriving the Schwinger-Dyson equation in Srednicki

Question

In Srednicki pg.136, he derives the Schwinger-dyson equation from:
$$
0=delta Z(J)=iint Dphi , e^{i[S+int d^4y, J_bphi_b]}int d^4x(frac{delta S}{delta phi_a}+J_a)deltaphi_a.tag{22.21}
$$
Then we apply $n$ functional derivatives wrt. $J$ on it, and we can get:
$$
begin{align}0=&int Dphi e^{iS}int d^4x[,ifrac{delta S}{delta phi_a}phi_{a_1}...phi_{a_n}cr&+sum_{j=1}^{n} phi_{a_1}(x_1)...delta_{aa_j}delta^4(x-x_j)...phi_{a_n}(x_j)]deltaphi_atag{22.22}
end{align}$$
after setting $J=0$. But I am stuck when I try to go through this procedure. For example, I simply do one functional derivatives:
$$
frac{delta}{delta J_{a_1}(x_1)}Z(J)=int Dphi,e^{i[S+int d^4x,J_aphi_a]}frac{delta}{delta J_{a_1}(x_1)}i(S+int d^4x J_aphi_a).
$$
And I am confused by the $S[phi]$ term, how can it become of the form $frac{delta S}{deltaphi_a}phi_{a}$? 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:
$$
frac{delta}{delta J_{a_1}(x_1)} delta Z(J) =int Dphi...[phi_{a_1}(x_1)int d^4x(frac{delta S}{deltaphi}+J)+int d^4xdelta_{aa_1}delta^4(x-x_1)]deltaphi.
$$

ohneVal · Answer

The first equation you wrote as Srednicki says, it is a variation with respect to $phi(x)$ not with respect to $J$, you are trying to get an expression related to symmetries when shifting $phirightarrow phi +deltaphi$. Thereafter the next functional derivatives are taken with respect to $J$ and then set $J=0$. You should compute
$$frac{delta Z}{delta phi(x')}$$
first.

Albin · Answer

To slightly expand upon the correct answers given by ohneVal and the OP, when one takes just one functional derivative of $delta Z(J)$ (obtained from first varying wrt $phi(x)$) but now wrt $J_{a_1}(x_1)$, there are two terms arising from the Leibniz rule
begin{align}
frac{delta}{delta J_{a_1}(x_1)}delta Z[J] &= i int mathcal{D}phi , e^{i left[S + int d^4y, J_bphi_bright]}left[iphi_{a_1}(x_1)int d^4x, left( frac{delta S}{delta phi_a(x)} + J_a(x)right)deltaphi_a(x) right.nonumber
&qquad{}left.+ int d^4x, delta_{aa_1}delta^4(x-x_1)deltaphi_a(x)right]nonumber
&=i int mathcal{D}phi , e^{i left[S + int d^4y, J_bphi_bright]}int d^4 x,left[i left( frac{delta S}{delta phi_a(x)} + J_a(x)right)phi_{a_1}(x_1) right.nonumber
&qquad{}+ delta_{aa_1}delta^4(x-x_1)bigg]deltaphi_a(x).
end{align}
Repeating this for $n$ values of $j$, and then setting $J_a(x) = 0$, we obtain the desired Eq. (22.22) in Srednicki.

Deriving the Schwinger-Dyson equation in Srednicki

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