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The Number e in Calculus

In calculus there is the notion of a derivative of a function, which is a measure of its rate of change with respect to changes in its input. When one differentiates an exponential function of the form f(x) = a^x one gets
                       f(x+h) - f(x)
        f'(x) =  lim   -------------
                 h->0       h
                        x + h    x
                       a      - a
              =  lim   -------------
                 h->0       h
                        x   h    x
                       a  a   - a
              =  lim   ------------
                 h->0       h
                 x        a  - 1
              = a   lim   ------
                    h->0    h

which is a^x times a number (the limit of (a^h-1)/h as h goes to 0) which is a constant independent of x, depending only on the base a of the exponential.

Therefore, the derivative of an exponential function is just a constant times the function value. What the constant is depends on what the base of the exponential function is.

The number e is that value of the base which yields the constant 1, so that the derivative of the function f(x) = e^x is actually equal to e^x itself.

Moreover, the derivative of the more general function f(x) = B e^(Rx) is R B e^(Rx). If we were using some other number a as the base instead of e, there would be an additional constant out front.

Not only does this make it more natural and convenient to use e as the base in exponentials and logarithms, but it relates back to the compound interest interpretation given earlier. If a bank account is growing under compound interest, with an interest rate of R per unit time, that means that at any instant in time the growth rate is R times the current balance. We've already seen that the balance at the end of the period is B e^R where B is the beginning balance. More generally, to figure out the balance somewhere in the middle of the time period, let t denote the fraction of the time period that has passed; it turns out that the balance at that time is B e^(Rt). The derivative of this function is B R e^(Rt) which is R times the function value; this is just another way of saying that the growth rate is R times the current balance.

This page last updated: September 1, 1997
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