Let us consider a functional determinant $$\det G^{-1}(x,y;g_{\mu\nu})$$ where the operator $G^{-1}(x,y;g_{\mu\nu})$ reads $$G^{-1}(x,y;g_{\mu\nu})=\delta^{(4)}(x-y)\sqrt{-g(y)}\left(g^{\mu\nu}(y)\nabla^{(y)}_\mu\nabla^{(y)}_\nu+m^2\right).$$ Such an operator appears in the one-loop effective action for a scalar field in curved spacetime (in the metric signature (+,-,-,-)). My question is how to do the following derivative $$\frac{\delta \log\det G^{-1}{(x,y;g_{\mu\nu})}}{\delta g^{\mu\nu}}?\tag{1}$$ I would think that such a calculation may show up in some textbooks on QFT in curved spacetime. So any recommendation on references is also greatly appreciated.
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Let me explain more why I found that doing the above derivative is difficult. In a scalar field theory in flat spacetime, we also have a term $$\log\det G^{-1}(x,y;\varphi)$$ with $$G^{-1}(x,y;\varphi)=\delta^{(4)}(x-y)\left(\eta^{\mu\nu}\partial^{(y)}_\mu\partial^{(y)}_\nu+V''(\varphi)\right)$$ in the effective action in presence of a scalar background $\varphi$. But now the derivative can be calculated as follows. $$\frac{\delta \log\det G^{-1}(x,y;\varphi)}{\delta\varphi}=\frac{\delta {\rm Tr}\log G^{-1}(x,y;\varphi)}{\delta\varphi}\\ =\int d^4 x d^4y\frac{\partial G^{-1}(x,y;\varphi)}{\partial\varphi}G(y,x;\varphi)=V'''(\varphi(x))G(x,x;\varphi)$$ where $G$ is the Green's function of the operator $G^{-1}$.
However, for the derivative (1), there are subtleties. First $$\frac{\partial G^{-1}(x,y;g_{\mu\nu})}{\partial g^{\mu\nu}}$$ is a differential operator. If one follows the above procedure, one should find $$\frac{\delta \log\det G^{-1}{(x,y;g_{\mu\nu})}}{\delta g^{\mu\nu}}\supset \int d^4x \left[\nabla^{(x)}_\mu\nabla^{(x)}_\nu G(x,y;g_{\rho\sigma})|_{y=x}\right]$$ which is divergent (recall $G$ is the Green's function and you have $\delta^{(4)}(0)$ in the above expression).
