For me these concept seem intuitively linked, simply because derivation measures the rate of change, and that compounding definition aims to find the compounding function with constant growth. The factorial definition only seems to make for an easier proof of the derivation of e^x.
That's kind of the point. The only background knowledge you need to understand my definition is how to differentiate a polynomial. Anyone capable of understanding what it even means to find a function that is its own derivative is almost certainly going to know that.
