I understand that you consider a "parametrized" identity on a space of functions $$ \mathcal{L}_{a,b}(f,g)=\[f-a\]\[g-b\]+a\[g-b\]+b\[f-a\]-fg+ab=0 $$ and two different linear forms acting on that space, namely $\phi : f\mapsto f'(x_0)$ and $\psi : f\mapsto\mathbb{E}(f)$, which induce the identities $\phi \[\mathcal{L}_{x_0,\;x_0}(f,g)\]=0$ (the product rule) and $\psi\[\mathcal{L}_{\mathbb{E}(f),\;\mathbb{E}(g)}(f,g)\]=0$ (the definition of the variance).
Of course you can evaluate this identity along all the (linear) maps you want playing with the parameters $a$ and $b$, but maybe your question was about a potential more subtle relation between "derivative" and "variance" of functions ? In this case, I have the impression that these two identities are different in nature, since the parameters $a$ and $b$ have to be chosen in a very different way for each form.