Beyond e^x

Prerequisites: The derivative of ex and the chain rule.

Goal: One can find the derivative of f(x)=ex by using the definition to write

and then use the calculator to guess that . Notice that is the derivative of f at zero.

In this laboratory we investigate the derivative of the more general situation: where the base, b, of the exponential function is not necessarily e. We start the same way:

This is a remarkable property of exponential functions. The derivative is just a constant times the original function.

In this laboratory we choose different values of b and zoom in on the graph of the function at x=0, until it looks essentially like a straight line, and measure the slope. As we continue to zoom in, the slopes get closer to , the derivative of f at 0. This value is the constant that we multiply times our original function to get the derivative for all x. In this laboratory we will also gather strong numerical evidence for a relation between this constant and the base b.

Begin: Start Maple, then copy and paste the following code into the Maple worksheet. The color is only for your own reference (Maple does not use purple). You may want to edit the purple characters, but probably not the red.

```
restart: with(plots):
a:= 0;
h:= 1;
b:= 2;
xMin := a-(3/2)*h:
xMax := a+(3/2)*h:
f := x -> b^x :
rise:=f(a+h)-f(a);
run:=h;
m:=rise/run;
Function:=plot(f(x),x=xMin..xMax,color=red,scaling=unconstrained);
Secant:=plot(f(a)+m*(x-a),x=xMin..xMax,linestyle=DASH,color=blue);
Triangle:=pointplot({[a,f(a)],[a+h,f(a)],[a+h,f(a+h)]},symbol=circle,color=black);
RiseLine:=pointplot({[a+h,f(a)],[a+h,f(a+h)]}, connect=true);
RunLine:=pointplot({[a,f(a)],[a+h,f(a)]}, connect=true);
display(Function,Secant,Triangle,RiseLine,RunLine);
m;
```

If you previously did the Derivatives by Zooming experiment, you may want to skip to step 4 below. If you did not do that experiment, or want to refresh your memory, then steps 1-3 will allow you to explore what the Maple code does when you execute it.

1. Copy and paste the code into Maple. After pasting the code into Maple, press [enter] (or [return]). After Maple executes the commands, you should see the graph of a function, a secant line through (a, f(a)) and (a+h, f(a+h)), and a triangle showing the rise and run from which the slope of the secant line is found. The value of the slope is displayed under the graph. Find and identify all these features.

2. The values in the Maple code in purple are parts that we will be editing. First, try changing h to 0.1 instead of 1. What changed? Similarly, change the value of a from 0 to 0.5. Experiment until you understand the roles of h and a.

3. Similarly experiment with the definition of the function. What happens if we change the function from bx to x3 ? To sin(x)? Experiment with a number of different functions.

4. When you are done experimenting, change the values of a, h, and the function back to their original values: a:= 0; h:= 1; f := x -> b^x;. What happens when you change the value of b? Experiment with several values of b. Can you find values where the graph is decreasing? Can you find values where the graph is horizontal?

5. Set b equal to 2. Zoom in on the point (0,1) by shrinking the values of h until you can tell at least the first three digits of the value the slopes are approaching. Try some very small window sizes such as .00001. Repeat for b equal to 3. Which value of b gives a slope at x=0 that is larger than 1? Which gives a value smaller than 1? Can you find a value of b whose slope at x=0 is equal to 1? Could you have predicted this value?

6. Enter the Maple line: ln(b); after the first > prompt that appears below the output from the previous work. What does Maple return when you execute the line? Now replace ln(b); with evalf(ln(b)); and see what Maple returns. ( In Maple, ln(b) stands for the exact value of the natural logarithm of b. evalf means find the floating point, i.e. decimal, approximate value of ln(b).) What looks familiar about the decimal value? Check with other values of b and see if the relation still holds. If so, make a conjecture about the relation of the slope of the tangent line to bx at x=0 and the value of b.

7. Use the results in the question above to conjecture what the right side of might be. (Hint: re-read the material at the top of this page to see how things fit together.)

8. Verify (prove) your conjecture by writing bx as bx=(eln(b)) x=e(ln(b)x) and taking the derivative using the derivative of ex and the chain rule. Explain.

Last Update: 2007/06/21