general relativity - Does it make sense to ask how the covariant derivative act on the partial derivative $nabla_mu ( partial_sigma)$? If so, what is the answer?

I want to find out how the covariant derivative acts on terms containing a partial derivative, e.g. $\nabla_\mu(k^\sigma\partial_\sigma l_\nu)$. But I don't know how to evaluate the terms of the form $\nabla_\mu(\partial_\sigma)$. If one writes

$$\nabla_\mu(k^\sigma\partial_\sigma l_\nu) = \nabla_\mu(g^{\rho \sigma}k_\rho\partial_\sigma l_\nu) = g^{\rho \sigma}\nabla_\mu(k_\rho\partial_\sigma l_\nu) = g^{\rho \sigma}\left[ \nabla_\mu(k_\rho)\partial_\sigma l_\nu + k_\rho\nabla_\mu(\partial_\sigma )l_\nu + k_\rho\partial_\sigma\nabla_\mu( l_\nu) \right]$$ Problem: How to determine $\nabla_\mu(\partial_\sigma )$. How do I work it out, and understand whatever the answer, that it makes sense? Have I made a mistake?

EDIT: I add the context: suppose $k^a$ and $l^a$ are killing vector. Then I want to prove that the commutator $[k,l]_\alpha = k^\sigma\partial_\sigma l_\alpha - l^\sigma\partial_\sigma k_\alpha$ is a Killing vector. If you write out $\nabla_{(\mu}[k,l]_{\nu)}$, then you find these terms immediately.

Answer

If you want to use this for commutators, then either consider

$$\nabla_\sigma([k,l]^\mu)=\partial_\sigma[k,l]^\mu+\Gamma^\mu_{\sigma\kappa}[k,l]^\kappa=\partial_\sigma(k^\nu\partial_\nu l^\mu-l^\nu\partial_\nu k^\mu)+\Gamma^\mu_{\sigma\kappa}(k^\nu\partial_\nu l^\kappa-l^\nu\partial_\nu k^\kappa)...$$

and use this for further calculations, or consider that for any torsionless connection, we have

$$[k,l]^\mu=k^\nu\partial_\nu l^\mu-l^\nu\partial_\nu k^\mu\equiv k^\nu\nabla_\nu l^\mu-l^\nu\nabla_\nu k^\mu,$$ and the latter expression contains only terms that are 'covariant'.