Basically I want to find a general algorithm that lets me, given any function y = f(x) (I'm not too worried about multivariable functions quite yet), I can come up with a matrix A so that x is mapped to f'(x). I want to do this so that I can sleep at night knowing that the solution to a linear ODE can be y_{gen}= y_{homogeneous}+ y_{particular }because derivatives are linear transforms. The moral of the story is that I want a solid understanding of why derivatives are linear transformations.

Sorry if this is a really stupid question to ask or pose.

Anthony