Derivative of a Constant Matrix

[Mandelbroth]

Let's suppose we have a matrix $\textbf{C}=\begin{bmatrix}c_{1,1} & c_{1,2} & c_{1,3} & \cdots & c_{1,m} \\ c_{2,1} & c_{2,2} & c_{2,3} & \cdots & c_{2,m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c_{n,1} & c_{n,2} & c_{n,3} & \cdots & c_{n,m}\end{bmatrix}$, such that $\forall i,j\in\mathbb{Z}:1 \leq i \leq n, 1 \leq j \leq m, \, \frac{\partial c_{i,j}}{\partial x}=0$. In other words, a constant matrix.

Is $\frac{d\textbf{C}}{dx}$ equal to 0 or the nxm 0-matrix? I thought it was the latter of the two, but I was thinking about it today and I wasn't sure.

 

[Ackbach]

Definitely the latter. Just think of the way differentiation works for vectors: component-wise. Then jack the dimension up by one to get matrices, and you'll see that it still works the same way.