So I know that since differentiation is linear, for many types of functions we can represent it using linear algebra. Famous examples include polynomials, if we represent them with their coefficients stored in vectors. Such a matrix performing differentiation on 4th order polynomial could look like:
$$D(4) = \left[\begin{array}{ccccc} 0&1&0&0&0\\ 0&0&2&0&0\\ 0&0&0&3&0\\ 0&0&0&0&4 \end{array}\right] \text{assuming order of coefficients stored as } \left[\begin{array}{ccccc} x^0\\ x^1\\ x^2\\ x^3\\ x^4 \end{array}\right]$$
Other families of functions have other differential operators, the trigonometric functions are well known for this:
$$D_{sintrig} = \left[\begin{array}{rr} 0&1\\ -1&0 \end{array}\right] \text{assuming coefficients stored like} \left[\begin{array}{rr} \sin\\ \cos \end{array}\right]$$ and the hyperbolic trigs: $$D_{hyptrig} = \left[\begin{array}{rr} 0&1\\ 1&0 \end{array}\right] \text{assuming coefficients stored like} \left[\begin{array}{rr} \sinh\\ \cosh \end{array}\right]$$ Then we can go one step further, investigating what happens when we use chain rule. A famous example for this is the Hermite polynomials which are the polynomials resulting in iteratively differentiating the function $e^{-x^2}$ which is of the form $\exp(P(x))$. This also gives linear representations, although more complicated.
So finally, to my question... do there exist systematic approaches to build such differential linear algebra "machinery" for even more complicated families of functions?