Why the exponential function?

In the first lecture of the point processes course, we have used at least twice the exponential function. It is maybe worth to shortly explain why the exponential function can be represented in different ways.
We are mainly concerned with the identities

e^{x} =  \sum_{n=0}^\infty \frac{x^n}{n!}= \lim_n (1+\frac{x}{n})^n 

Let us start by defining the exponential function to be the only function for which

f(x) = f'(x)

up to a scalar factor. So, we have to check that both definitions enjoy this property. If it is case, then they will both agree with the exponential function, assumed that everything converges.

Let us check it for the first formula. Deriving term by term yields

d/dx\sum_{n=0}^\infty \frac{x^n}{n!} = \sum_{n=1}^\infty \frac{n x^{n-1}}{n!}=\sum_{n=1}^\infty \frac{ x^{n-1}}{(n-1)!}

By calling k the term n-1 we obtain our desired identity.

For the second one, observe that

d/dx f^n(x) = nf'(x)f^{n-1}(x) 

Applying this to the product in the formula we obtain

d/dx(1+\frac{x}{n})^n  =\frac{n}{n} (1+\frac{x}{n})^{n-1}

But since

\lim_n (1+\frac{x}{n})=1

one could hope that the identity holds. But here we have some difficulties with the convergence, indeed.