QFT - Generating Functional 
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*The problem statement, all variables and given/known data


Hi
I am looking at the attached question part c)



*Relevant equations


below


*The attempt at a solution


so if i take $\frac{\partial^{(n-1)}}{\partial_{(n-1)}} $  of (2) it is clear I can get the $\frac{i}{h} (\lambda_2 +\lambda_4 )$ like-term, but I am unsure about the $nG_{n-1}$ .
There's obviously no other derivatives on the RHS so I will only yield a $G_{n-1}$ and that looks fine, I am a bit confused though, I can yield this from the $ Z[J] $ alone on the RHS, whereas the RHS is $Z[J]$ 'multiplied by' (it is already inside the integral) the extra term of $S'[\Phi] + J$ . So I suspect this extra term is the reason we get the $n$ factor but I am unsure how.
Looking at the LHS there is a single $J$ so it looks like this gives a factor of $1$ and then we take across $(n-1)$ from the RHS.
If I take a derivative wrt $J$, on the LHS I can either act on the exponential or the single $J$ (but can only act on this $J$ once,) on the RHS it's the same story, with the difference that on the LHS the $J$ is outside the integral but on the RHS it is inside the integral, I'm trying to use this to deduce where the factor of $n$ comes from but I am struggling..

a) You will find it easier to work in Euclidean space so the integrand becomes $Exp(S(\phi)/\hbar + J \phi)$ (your action is negative) , which falls off rapidly with $\phi$ and also gets the proper vacuum state.
b) The right side of eq. 2 is the integral of a total derivative, so it is 0.
c) The n comes from $\partial^n/\partial J^n (J exp(J \phi))$ on the left side.
