# Derivative of an expectation function

For a general function $$g$$, define $$f=\mathbb{E}_{(x,y)\sim \mathcal{N}(0, \Sigma)}[g(x)g(y)]$$, where $$\Sigma=\begin{pmatrix}a & c \\c & b \end{pmatrix}$$, and $$a>0,b>0,ab>c^2$$.

I am looking for a Mathematica solution for finding the partial derivative with respect to $$c$$ and evaluate it at the point $$(1,1,0)$$.

The solution by hand is $$\frac{\partial f}{\partial c}|_{a=1,b=1,c=0}(\mathbb{E}_{z\sim\mathcal{N}(0,1)}[zg(z)])^2.$$

I've tried the code:

gg = Function[{a, b, c},
Expectation[
g[m] g[n], {m, n} \[Distributed]
MultinormalDistribution[{0, 0}, {{a, c}, {c, b}}]]]

Derivative[0, 0, 1][gg][1, 1, 0]


but I got something that really confuses me. Thanks for your help in advance.