Let $\nabla$ be levi civita connection on Riemannian manifold $M$. I was wondering, what is $\nabla_{\alpha}(\partial_{\beta}g_{\mu\nu})$?
Is it $\partial_{\alpha}\partial_{\beta}g_{\mu \nu}-\Gamma^{\sigma}_{\alpha \beta}\partial_{\sigma}g_{\mu\nu}-\Gamma^{\sigma}_{\alpha \nu}\partial_{\beta}g_{\sigma \nu}-\Gamma^{\sigma}_{\alpha \nu}\partial_{\beta}g_{\mu \sigma}$?
This is a weird expression to me, because $\partial_{a}g_{bc}$ is not a tensor, and so applying a covariant derivative to it is not natural. Generally, one would evaluate double covariant derivatives by going from "outside" in, so that you are are always evaluating tensors. So, for example, for some tensor $T_{ab}$, you have:
$$ \nabla_{c}\nabla_{d}T_{ab} = \partial_{c}\left(\nabla_{d}T_{ab}\right) - \Gamma_{cd}{}^{e}\nabla_{e}T_{ab} - \Gamma_{ca}{}^{e}\nabla_{d}T_{eb} - \Gamma_{cb}{}^{e}\nabla_{e}T_{ae}$$
And you then expand out the rest with the ordinary expression for $\nabla_{a}T_{bc}$ and applying the ordinary product rule for partial derivatives
