If $M$ is your matrix, then it represents a linear $f\colon \mathbb{R}^n \to \mathbb{R}^n$, thus when you do $M(T)$ by row times column multiplication you obtain a vectorial expression for your $f(T)$. Thus $\frac{\partial M}{\partial T}$ is just the derivative of the vector $MT$, which you do component-wise. Were you looking for something different?
EDIT: I actually see now that you most likely have a vector space of functions, but this doesn't change much at all: see that if $T = (f_1(t),f_2(t))^T$ and $M$ represents a linear homomorphism $F\colon V \to V$, then you have that $\frac{dF}{dt}(f_1(t),f_2(t))^T$ is actually $F(\frac{df_1(t)}{dt}, \frac{df_2(t)}{dt})$. This is actually straight forward to see: just compute $MT$ by row $\times$ column multiplication and then derive with respect to $t$.