Derive Definition of Exponential Function (Euler's Number) from Compound Interest
This example demonstrates how the concept of compound interest can be used to derive the exponential function by calculating a numeric value for Euler’s Number. Note, while this is a valid definition of the function, this site considers purely as shorthand for and instead defines the exponential function using a power series.
Start with the formula for compound interest, shown below. This formula models an exponential growth curve that is dependent the principle amount , rate of interest , the frequency of investment and the time elapsed .
The question that leads to the exponential function is “what happens when the interest compounds continuously?” In other words, what happens when approaches infinity. This can be answer using the limit notation, shown in the equation below.
The rest of this step discusses how one would formulate this question. Otherwise, skip to step and
Having a limit in the definition of the function poses a couple challenges, so it is in our best interest to manipulate the expression into a more useful form. Observe that is a scalar value and can be moved outside the limit.
Next, observe the growth rate prevents us from evaluating the limit. To solve this, perform a substitution where , leaving us with a limit which is only dependent on the variable .
Then we can apply a property of exponentiation which allows the variables to be split apart. This gives the expression below.
Finally, we have a limit that can be evaluated to approximate a numeric value. The value of this limit, as it turns out, is equal to the numeric value of Euler’s Number. This is shown below.
Substituting for this value back into the original equation gives us the formula below, which is the formula for population growth.
To finish this example, we are are going to make the formula more abstract by replacing the variable with , setting the principle amount to and the growth rate constant . This leaves us with the exponential function.