I am trying to compute the $2$-point Green function $\tau_2(x,y)$ for free Dirac fields. The corresponding formula for $\tau_2(x,y)$ is given by
$$\tau_2(x,y) = -\frac{\delta^2}{\delta\eta_x \delta \bar{\eta}_y} \, Z_0[\eta_w, \bar{\eta}_z],$$
where $Z_0$ is the generating functional for free Dirac fields given by $$ Z_0[\eta_w, \bar{\eta}_z] = \exp\left(-i\int \bar{\eta}_z \, S(z-w) \, \eta_w \, dz \, dw\right). $$
Here, $\eta$ and $\bar{\eta}$ are source terms. Also, $S^{-1} = i\gamma\cdot\partial - m$ is the operator appearing in the quadratic term of the lagrangian.
Notation $\eta_x \equiv \eta(x)$ etc.
At first, I determine $\frac{\delta Z_0}{\delta\bar{\eta}_y}$ as
$$ \frac{\delta Z_0}{\delta\bar{\eta}_y} = -iZ_0 \int S(y-w) \eta_w \, dw. \label{a}\tag{1} $$
Then I try to compute
\begin{align} -\frac{\delta^2 Z_0}{\delta\eta_x\delta\bar{\eta}_y} &= -\frac{\delta}{\delta\eta_x} \left[-iZ_0 \int S(y-w) \eta_w \, dw \right] \\ &= i\frac{\delta}{\delta\eta_x} \left[Z_0[\eta_w, \bar{\eta}_z] \int S(y-w) \eta_w \, dw \right] \label{b}\tag{2} \end{align}
Question
How to proceed from this step (eq. (\ref{b}))? I have to take the functional derivative of the product of two Grassmann functionals. What is the relevant formula for it? If you also mention any reference, that would be great.
In eq. (\ref{a}) I have written $Z_0$ before the functional derivative part. Should I write it after the functional derivative term? In other words, what is the chain rule for Grassmann functionals?
In the Appendix, I have mentioned the formula to take the functional derivative of a product of Grassmann functionals. Let's say, I have a product of some Grassmann functionals and an ordinary function $f(x) \in \mathbb{C} \forall x$. Then how to evaluate this functional derivative? That is,
$$ \frac{\delta}{\delta\psi(x)} [\psi(y_1) f(y_2) \psi(y_3)] = ? $$ where $\psi$ is a Grassmann field.
Appendix
The formula used to compute the eq. (\ref{a}) is given below.
$$ \frac{\delta}{\delta\psi(x)} [\psi(y_1) \cdots \psi(y_n)] = \delta(y_1-x) \psi(y_2) \cdots \psi(y_n) + (-1) \delta(y_2-x) \psi(y_1) \cdots \psi(y_n) + \cdots \cdots + (-1)^{n-1} \delta(y_n-x) \psi(y_1) \cdots \psi(y_{n-1}). $$