The function f(x)=ex is quite peculiar: it is the only function (up to
scale) whose derivative is itself. We have
f′(x)=limh→0f(x+h)−f(x)h=limh→0ex+h−exh=limh→0exeh−exh=ex⋅(limh→0eh−1h).
The last step was made possible by the fact that ex doesn't depend on h. Notice that
limh→0eh−1h=f′(0),
which we have defined to be 1 earlier. This implies that the derivative of ex is ex!