I know that for two k-vectors, say A and B, ∂A/∂B would be a square k×k matrix whose (i,j)-th element would be ∂Ai/∂Bj.
But could someone please explain how the partial derivative look like if we were differentiating k×k matrix instead? That is, M is a k×k matrix, x is a k-vector, how can we write ∂M/∂x?