In quantum field theoy, let's say the free scalar field theory, we can compute the $n$-point correlation function $\langle 0|T(\phi(x_1),...,\phi(x_n))|0\rangle$ using Wick's theorem.
Alternatively, we can differentiate the generating the generating function $$W_0[J]=\exp\bigg[-\frac{1}{2}\frac{i}{\hbar}\int d^4x'\,d^4x\,\,J(x')\Delta_F(x'-x)J(x)\bigg]$$ which gives $$\langle 0|T(\phi(x_1),...,\phi(x_n))|0\rangle=\bigg(\frac{\hbar}{i}\bigg)^n\frac{\delta^nW[J]}{\delta J(x_1)...\delta J(x_n)}\bigg|_{J=0}.$$
My question is: the path integral derivation of $W_0[J]$ I found in standard textbooks requires some extraordinary leap of faith (with fermions the leap only gets bigger), so it is not immediately obvious to me why the two approaches must yield identical answers.
Of course we can check the special case of a free scalar field theory, as it's just a combinatorial problem. However, it's there a better explanation why this should hold? Not just free scalar field, but free fermions, Yukawa theory, QED etc?
