For simplicity, suppose I have a $3\times3$ matrix $M$ and two vectors $u,v$ of size $3$. Let $F(M,u)=Mu$. I know that \begin{equation} \frac{\partial F(M,u)}{\partial u}=\frac{\partial Mu}{\partial u}=M. \end{equation} Then what the result of \begin{equation} \frac{\partial F(M,u)}{\partial M}=\frac{\partial Mu}{\partial M} \end{equation} Is there a special name for it? And how do I compute the following equation \begin{equation} \frac{\partial Mu}{\partial M}v \end{equation}
Basically, my second question is equivalent how to do multiplication between a $3\times3\times3$ matrix and a vector.