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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
one gets
which is times a number (the limit of
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
is actually equal to itself.

Moreover, the derivative of the more general function is . 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 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
.
The derivative of this function is 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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Philip Spencer

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