It doesn't appear miraculous because knowing all derivatives for a function is quite a lot of information about a function. A polynomial of degree six can be described with six derivatives but it can also be described with six factors. There is no information advantage.
If you define a polynomial by its coefficients, and then take its derivatives at a point and forget the coefficients, it would be reasonable to wonder whether you had lost some information. The miracle is that you haven't (if you also remember that the original function is a polynomial).
