derivation of the formula for continuously-compounded interest

Here is the link to such a derivation.
It gives what we might call the ‘bottom-up’ derivation.
(Another option is the ‘top-down’ derivation, in which n going to infinity is replaced by 1/x with x going to 0, and using the exponential-logarithmic definition of a^b, and considering the behavior of the derivative that arises from that.)
keywords: Calculus, proof, money, calculation, limit