Its Own Derivative

The function eˣ has a remarkable property: its derivative equals itself. The rate of change of eˣ at any point equals the value of eˣ at that same point. (Any multiple like 2eˣ shares this trait, since its derivative is 2eˣ; what singles out eˣ is that it also equals 1 at x = 0.)

That property makes eˣ the natural solution to any equation describing growth that speeds up in proportion to its current size.

Bacterial colonies double at rates governed by e. Radioactive atoms decay with e. Newton's law of cooling uses e. In finance, continuously compounded interest uses the formula A = Peʳᵗ, where P is principal, r is rate, and t is time. The number e is the heartbeat of exponential processes.