Let $\nabla$ be levi civita connection on Riemannian manifold $M$. I was wondering, what is $\nabla_{\alpha}(\partial_{\beta}g_{\mu\nu})$?
Is it $\partial_{\alpha}\partial_{\beta}g_{\mu \nu}-\Gamma^{\sigma}_{\alpha \beta}\partial_{\sigma}g_{\mu\nu}-\Gamma^{\sigma}_{\alpha \nu}\partial_{\beta}g_{\sigma \nu}-\Gamma^{\sigma}_{\alpha \nu}\partial_{\beta}g_{\mu \sigma}$?
Some hints.
Recall the definition of covariant derivative:
$$\nabla_{\mu}g_{\alpha\beta} \equiv \dfrac{\partial g_{\alpha\beta}}{\partial x^{\mu}} - \Gamma^{\eta}_{\mu\alpha}g_{\eta\beta} - \Gamma^{\gamma}_{\mu\beta}g_{\alpha\gamma}$$
As well as the definition of Christoffel Symbols
$$\Gamma^{\ell}_{ij} \equiv \dfrac{1}{2}g^{\ell k}\left(\dfrac{\partial g_{\ell i}}{\partial x^{j}} + \dfrac{\partial g_{\ell j}}{\partial x^{i}} - \dfrac{\partial g_{ij}}{\partial x^{\ell}}\right)$$
Can you then manage to derive the form of $\nabla_a(\partial_{\beta} g_{\mu\nu})$?