In Chapter 2 you solved the derivative using the Difference
Quotient for the function y = f(x). You will do
the same thing in this chapter for the exponential function b^x.
While doing so, you will sort of 'stumble' upon one of the most
profound items in all of mathematics; that is the fact that the
derivative of e^x is equal to itself. This is the only
function with this property (except for the trivial function f(x)
= 0), and by studying it, you will develop much of the theory
for all exponential and logarithmic functions.
As with most concepts in this book, looking at a graphical
representation is the best way of starting out. The graph of b^x
is interesting because of the fact that, by changing the base
b, the general shape of the graph does not change. The
question you want answered is whether this similarity also holds
true for its derivative.
By plotting the function b^x and its Difference Quotient in the same Graph Theory you will be able to use LiveMath's tools to show a dramatic relationship. The reason you use the Difference Quotient rather than just solving the derivative with LiveMath's derivative Op, will become evident as you go on from there.
As a refresher, the difference quotient for b^x is shown below. Remember to put an (x + h) everywhere there is an xin the function to determine f (x + h).
Example 4.1 The DQ of the Exponential
Function
LiveMath Internet: The Exponential Function
The common phrase, "the growth is exponential" as a description of radical growth, is really brought home when you study these plots. As you can readily see, very small changes in x return very large changes in y.
Click on the LiveMath Internet link below, change the value
for b and observe the derivative plot as it relates to
the exponential function. Use values between 2 and 10 for b.
LiveMath Internet: The Exponential Function
The QuickTime movie below is the same plot animated. Drag the pointer back and forth watching the derivative plot (red) start underneith the function and end above the function as b goes from 2 to 4.75. When the value of b is at 2.75, the plots seem to be on top of each other.
This demonstration leads to the remarkable possibility that the exponential function, to a base near 2.7, may have a derivative that is equal to itself! At some value b, the derivative function you have defined with the Difference Quotient, is equal to (b^x).
The discussion that follows has subtle, but profound meaning, so read carefully. The revelation upon its understanding will make the whole process worth the effort!
As stated earlier, the reason you used the Difference Quotient rather than having LiveMath just generate the Derivative for you, has a purpose. The Difference Quotient for the Exponential Function has interesting characteristics that will give you the key to the understanding of the function e^x.
The Difference Quotient for the function b^x is
given again below which, when the value of h approaches
0, becomes the Derivative. This is stated mathematically
as the limit as h->0 and is shown below:
Although not readily apparent, a very important characteristic of this expression is the fact that you can isolate the original function b^x from a constant expression within the Difference Quotient! Using LiveMath, you can manipulate the above expression to determine that constant.
Example 4.2 Solving the DQ of the Exponential Function
Placing the expression back into the form of a limit, you can pull out b^x making the limit look like the graphic below.
and you can look at this limit in the following manner.
This means that the derivative of the Exponential Function is just the function itself times a constant. It is a constant because for any given base (b) the expression labeled constant above, does not depend on x. It depends on a value b, but is a constant as far as x is concerned.
If you isolate this expression and try different values for b you can look at the constants generated and make some observations. In the next example, you will discover a very interesting fact about this constant as it relates to values for b.
Example 4.3 The Exponential Function's Constant
The variable here is b and it has a domain of 2 to 4 which generates values which range from 0.6931712 for b = 2 to 1.3863905 for b = 4.
Somewhere in the range of b= 2.5 to b= 2.75 there looks to be a value that is equal to 1. If you can find this value, you have the base in which its derivative is equal to itself. This is because:
If the constant is equal to 1, then the derivative is equal to itself!
In the next example, you equate the constant expression to 1 and then solve for b.
Example 4.4 Which b is the Right b?
- Input the constant expression and make it equal to 1.
- Select and Isolate b (NOTE: you must have Auto-Casing turned OFF).
Where have you seen this expression before?
You know that you need to make h small, but for academic purposes, give h some values approaching .0001. Try using LiveMath Internet to choose some values. Below are some results.
You can see that after h gets small, the value for b approaches the value e (hence the reason e is calculated in the last Prop, in the LiveMath Internet example)! So what you have discovered is that the Exponential Function with a base of e has a derivative equal to itself! It is summarized with the graphic below.
The question still remains; what is the formula for the derivative of the general Exponential Function? You know that for any Exponential Function, the Difference Quotient states that the derivative is the original function times a constant. You have discovered that if the base is equal to e the constant is1, but how do you determine the constant for any other base b? Is there a short-cut to finding these? There is, and again, you can use LiveMath to find it
In Chapter #1 the phrase "to exponentiate" was introduced. It was shown that the inverse of the Natural Exponential Function is the Natural Log Function. If, like the method used in Example #1.17, you take an exponential argument and use it inside a log function, you create an identity which 'undoes' the argument ,and visa-versa.
This is confusing language, so look at the following graphic for help.
You can define the Exponential Function b^x as e taken to the power of x times the Natural Log Function. The reason this is true comes from the 3rd Property of Logs; ln(u^c) = c·ln(u) and the 4rd Law of Exponents; b^(xy) = (b^x)^y. Below is shown a little trick which will help you find the constant rule and allow you to do several proofs.
The last statement is true because the Natural Exponential Function is the inverse of the Natural Log Function.
Using this definition for b^x will allow you to finally find the derivative of the Exponential Function.
Example 4.5 The Derivative of the Exponential Function
Work ONLY on the RHS of the equation from now on.
selection
selection
So the derivative of the General Exponential Function is:
Now you know where all those numbers came from in Example 4.3. They are merely the Natural Log of the base (b). Back in Chapter 1 you learned that ln(e) = 1 and, of course, this confirms the discussion above where you determined, with the Difference Quotient, that the constant for the derivative of e was equal to 1.
The following summarizes what you have learned thus far in the chapter.
