dg.differential geometry – covariant derivative commute with Hodge star


Suppose $(M,g)$ is a Riemannian manifold. Let $nabla $ be the covariant derivative associated with the Levi-Civita connection. Suppose $*$ is the Hodge operator with respect to the metric $g$ on $Omega^p(M)$. If $omegain Omega^p(M)$ and $Xin T_pM$, do we have
$$*(nabla_X omega)=nabla_X (*omega)quad?$$