Constant Multiple

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.

What happens to the derivative of a function if we multiple the function by some constant?

Try the following:

  1. The example shows the line f (x) = kx. We know that the slope for the line f (x) = x is just 1. What does the k do to the slope? Move the k slider and you notice that the new slope is just k.

  2. Select the second example, showing a parabola multipled by k. When k > 1, what happens to the shape of the parabola? Move the k slider to see. What happens to the derivative? Set x = 1 and k = 1 and notice the value of f '(1). Now, change k to 2; what happens to the derivative at x = 1? Change k to other values, and see if you can detect a pattern in what happens to the derivative.

  3. Select the third example, showing a sine curve multiplied by a constant. What does changing k do to the derivative?

  4. Hopefully you have noticed that multiplying a function by a constant just multiplies the derivative by the same constant, or (d/dx)k*f(x)=k*f'(x).

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© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

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Citation: tdowney. (2009, August 06). Constant Multiple. Retrieved July 23, 2014, from Notre Dame OpenCourseWare Web site: http://ocw.nd.edu/mathematics/calculus-ii-for-business/calculus-applets-website/Constant%20Multiple.html.
Copyright 2012, Thomas S. Downey. This work is licensed under a Creative Commons Attribution 3.0 License. Creative Commons Attribution 3.0 License