Okay let's begin. Your question is actually the motivation for defining exp in the first place! If you have some function exp on C such that exp'(x) = exp(x) for any x in C, then for any variable y varying with parameter x in C such that dy/dx = y at every point, we have (at every point) d(y·exp(−x)/dx = dy/dx·exp(−x) − y·exp(−x) = 0, and hence (by Rolle's theorem equivalent) y·exp(−x) = c for some constant c. Thus y = c·exp(x).
