Exponents and logarithms: inverse functions

Derivatives of Exponentials

On this page we will calculate the slope of the exponential functions $ f_a(x) = a^x $ that we described earlier. This produces a startling result about the rate at which this function increases.

The Calculation

To calculate the slope, we will consider an interval between $
x $ and $ x + h $ , compute the ratio of average rate of change of the exponential function over this interval and then see what happens as the interval becomes very small.
The calculation begins like this:
$\frac{d a^x}{dx} = \lim_{h \to 0} \frac{a^{x+h}-a^x} {h}$

Now we will recall that $a^{x+h}=a^x a^h$ so that we can write
$\begin{eqnarray*} \frac{d a^x}{dx} & = & \lim_{h \to 0} \frac{\left(a^x a^h - a^x\right)}{h} \\ & = & \lim_{h\to 0} a^x \frac{a^h -1}{h} \\ & = & a^x \lim_{h\to 0} \frac{a^h - 1}{h} \end{eqnarray*}$
This last line is especially convenient because it separates the part of the calculation which depends on $ x $ and the part which depends on $ h $ . Notice that the part which depends on is just the original value of the function $ a^x
$ . The other part which depends on $ h $ is less clear cut---we have not seen a limit like this before. Let's begin to investigate it by setting $ x = 0 $ in the expression:

$\frac{d a^x}{dx}|_{x=0} = a^0 \lim_{h\to 0} \frac{a^h - 1}{h} = \lim_{h\to 0} \frac{a^h - 1}{h}$
This means that the limit is simply measuring the derivative of the function $ a^x $ at $ x= 0 $ . In other words,
$\frac{d a^x}{dx} = \frac{d a^x}{dx}|_{x=0} a^x$
Let's think about this for just a minute because it is telling us something very important. The derivative at $ x = 0 $ is just a number which we could denote by $ C(a) $ . Doing so allows us to write the equation as
$\frac{d a^x}{dx} = C(a) a^x$
To express this in words, we could say that the rate of growth of the function $ a^x $ is proportional to the function itself.

For your consideration:

Why can we not simply plug in the value h=0 into the expression $\lim_{h \to 0} \frac{\left(a^{h}-1 \right)}{h}$ to determine its limit? What happens to the numerator? The denominator?
Explain the statement "rate of growth of the function $ a^x $ is proportional to the function itself" in terms of what we've already learned about exponentials.
In the demonstrations below, you will see the graphs of $
2^x $ and $ 10^x $ . Use the zoom feature to estimate roughly the constants $ C(2) $ and $ C(10) $ .

Choosing a convenient base

We are now in position to see why a particular choice of base seems most convenient in calculus. For the function $ y=2^x $ , you will have found the derivative
$\frac{d}{dx} 2^x = C 2^x$
where $C \approx 0.693$ . This means that taking the derivative of results in the original function multiplied by a funny constant. (Similarly, the derivative of $ y=10^x $ is $\frac{d}{dx} 10^x = C\cdot 10^x$ , but with the constant C=2.303).
We will now define a number, which we call $ e $ , by requiring that $ C(e) = 1 $ . This gives the tidy result that

$\frac{d e^x}{dx}= e^x$
But how can we understand the value of $ e $ ? Well, it is defined as the number which has the property that $ e^x $ has slope 1 at $ x = 0 $ . Notice in the demonstration below that as we change the base a , the steepness of the graph changes.

Now we are asking if we can choose the base so that the slope at x = 0 is 1. Since the tangent line at $ x
= 0 $ must also pass through $ (0,1) $ , the tangent line should have the equation $ y = x + 1 $ . In the demonstration is shown the graph of an exponential and the line . We would like to vary the exponential so that the graph has the line as its tangent at $ x = 0 $ .

After playing around with this demonstration, you may believe that $e \cong 2.71828...$ . The number $ e $ is similar, in some ways, to $\pi$ : it is given to us by nature, we have no choice in the matter, and it is irrational (that is, it is not the ratio of two integers).
We have found some rather interesting facts here, and it is best to summarize them:

Observations

The derivative of any of the exponential functions is proportional to the original function.
If we choose e = 2.7182.. as the base, the constant of proportionality is 1.
The derivative of $ e^x $ is $ e^x $ .
The rate of increase of an exponential process is itself an exponential process. The bigger the size (of, say, the population, the faster it increases, etc.)