In the above applet, there is a pull-down menu at the top to select which function you would like to explore.
The selected function is plotted in the left window and its derivative on the right.
1. Exponential function with a fixed base
The initial example shows an exponential function with a base of k, a constant (initially 5 in the example). What does the derivative look like? It sort of looks like the original exponential function, but rising more steeply. Move the k slider around and notice what happens to the shape of the derivative.
Are there some values of k for which the derivative rises less steeply than the original curve? What value of k makes the two curves look similar? You can get even closer to this magic value for k by setting x = 1 and then watching the value of f '(1) (shown in a box in the right hand graph) as you move the k slider. Since f (1) = k, when f ' (1) = k, the two curves are identical. Once you get close using the k slider, you can also fine tune the value of k using the left and right arrow keys on your keyboard. You should find that for k ≈ 2.718 the function and its derivative are the same. The exact answer is k = e. In fact, you can type "e" into the k input box to make the curves the same. So,
What about when the base is a number other than e? It appears that the derivative is like the original exponential, but stretched or squished. In fact, that is what happens, and the shortcut is
2. Exponential function with a base between 0 and 1
What about for 0 < k < 1? Select the second example from the drop down menu. This is the same function, but now the k slider will let you select values from 0 to 5 (instead of just from 1 to 5, as in the previous example). What happens to the derivative curve? Why? What is the sign of the logarithm of a number between 0 and 1? The rule given above still works in this case.