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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.

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