---

The Definition of e and the Derivative of ex

---

Details

The calculation of the derivative of f(x) = a^x, for a > 0, is similar to that of 2^x and 3^x:

f ' (x) =

lim h → 0

a x + h − ax h

=  ax

lim h → 0

a h − 1 h

The quantity:

lim h → 0

a h − 1 h

doesn't depend on x, and so is a constant for any particular a. Therefore the derivative is proportional to the original function, with constant of proportionality:

lim h → 0

a h − 1 h

A calculator can not be used to evaluate this limit without knowing the value of a. However, when a = 2, the limit of 0.6931 is less than 1, and the derivative is smaller than the original function. When a = 3, the limit of 1.0986 is more than 1, and the derivative is greater than the original function. Is there an in-between case, when derivative and function are exactly equal? In other words, is there a value of a that makes:

d dx	(ax ) = ax   ?

If so, a function is found with a remarkable property that is equal to its own derivative. An a is to found such that:

lim h → 0

a h − 1 h

= 1

or, for small h:

lim h → 0

a h − 1 h

≈ 1

Solving for a suggests that a can be calculated as follows:

a ^h − 1 ≈ h

or:

a ^h ≈ 1 + h

so:

a ≈ (1 + h)^{1/ h}

Taking small values of h, as shown in the below table, it can be seen that a ≈ 2.718..., which looks like the number e.

h	(1 + h)1/h
0.001	2.716923932
0.0001	2.718145927
0.00001	2.718268237
0.000001	2.718280469
0.0000001	2.718281694
0.00000001	2.718281798

In fact, it can be shown that:

e =

lim h → 0

(1 + h)1/ h = 2.718...

and:

lim h → 0

e h − 1 h

= 1

This means that e^x is its own derivative:

d dx	(ex ) = ex

It turns out that the constants involved in the derivatives of 2^x and 3^x are natural logarithms. In fact, since 0.6931 ≈ ln 2 and 1.0986 ≈ ln 3, it is correct in stating that:

d dx	(2x ) = (ln 2)2x

and:

d dx	(3x ) = (ln 3)3x

In general, it can be stated that:

d dx	(ax ) = (ln a)ax

Fig1 shows the graph of the derivative of 2^x below the graph of the function. Fig2 shows the graph of the derivative of 3^x above the graph of the function. With e ≈ 2.718, the function e^x and its derivative are identical as shown in Fig3.


Fig1. Note that f(x) is above f '(x)


Fig2. It is hard to tell, but g(x) is below g '(x)


Fig3. Note that e^x is the same as its derivative, therefore the two functions lay over each other.

Since ln a is a constant, the derivative of a^x is proportional to a^x. Many quantities have rates of change which are proportional to themselves; for example, the simplest model of population growth has this property. The fact that the constant of proportionality is 1 when a = e makes e a particularly useful base for exponential functions.

Also, in mathematical nomenclature note that exp(x) is the same as e^x