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 .

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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 28, 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.
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