This file is part of a distribution of the Calculus Applets website (http://www.calculusapplets.com) (v1.1) which has been reformatted for the needs of this OpenCourseware course.

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.

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© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

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Citation: tdowney. (2009, August 06). Exponential Functions. Retrieved October 21, 2014, from Notre Dame OpenCourseWare Web site: http://ocw.nd.edu/mathematics/elements-of-calculus-i/calculus-applets-website/Exponential%20Functions.html.

Copyright 2012,
Thomas S. Downey.
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