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:

- 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*.

- 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.

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

- Hopefully you have noticed that multiplying a function by a constant just multiplies the derivative by the same constant, or .

For more information on rights and downloading, refer to http://www.calculusapplets.com/download.html.

© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

You must cause any files that you modify to carry prominent notices stating that you changed the files and the date of any change, and modified files must be put into a Java package different from edu.hws.

THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.

Citation: tdowney. (2009, August 06). Constant Multiple. Retrieved April 19, 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.