So here is the connection with differential equations:
The exponential function is the unique solution to the following differential equation:
The derivative of Y of X, with respect to X, equals Y of X, for all X,
and the initial value, Y of zero, equals one. This is about as simple as it can get with differential equations.
We're looking for a function, Y, that is its own derivative, and starts with a value of one.
It's clear now that the exponential function is a solution to that differential equation,
but as most other differential equations that are of interest, also this differential equation is so well-behaved
that the exponential function is the unique solution of this differential equation.