The term "matrix derivative" is a bit loaded - you can either mean the derivative of functions with matrix arguments, or functions with vector arguments that have some matrix multiplication terms. Either way, I don't really understand what the confusion is about - if you slightly modify the definition of a derivative to be directional (e.g. lim h->0 (f(X + hA) - f(X))/h) then all of this stuff looks the same (vector derivatives, matrix derivatives and so forth). Taking this perspective was very useful during my PhD where I had to work with analytic operator valued functions.
There’s a little more to it than that. Matrices have several properties like being symmetric or unitary (maybe even diagonal) that scalars don’t. Those enable rewritings to make systems more stable or computationally efficient.
That bears no relation to the symbolic differentiation in the OP though.
Even plain old multiplication and division, and even addition and subtraction have stability and efficiency problems on floats, which don't appear in symbolic solvers.
