It is’t the exponential function that’s a miracle, it’s the Maclaurin series. Think about it. The Maclaurin series says that in a very real sense, the value of a function at zero, and the value of all of its derivatives at zero, contains all of the information there is to know about it. Intuitively, this should be completely unexpected. Why in the world would the derivatives of a function evaluated at a single point tell you anything whatsoever about the value of the function arbitrarily far away from that point?
EDIT: as mentioned below, this is not true of all functions, even all functions that are infinitely differentiable at zero. But it is true for very large classes of functions.
EDIT: as mentioned below, this is not true of all functions, even all functions that are infinitely differentiable at zero. But it is true for very large classes of functions.