Exponential functions are somewhat special, in that their derivatives look a lot like the original function, as you have seen in previous examples.
Try the following:
- 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 anwer 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 .
- 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.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.