Gradient of matrix-vector product

Is there a way to make the identity of a gradient of a product of matrix and vector, similar to divergence identity, that would go something like this:

$\nabla(\textbf{M}.\textbf{c})= \nabla(\textbf{M}).\textbf{c}\ +\ ... (\text{not necessarily like this})$,

where M is a $n\times n$ matrix and $c$ is a $n\times 1$ matrix (column vector)?