
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 wellbehaved

that the exponential function is the unique solution of this differential equation.