# Hyperbolic Functions

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.

We define three hyperbolic functions as follows: , , and .The applet shows the graphs of these functions and their derivatives.

Try the following:

1. The applet initially shows the graph of cosh(x) on the left and its derivative on the right. The hyperbolic cosine looks sort of like a parabola, but looking at the derivative (which for a parabola is a straight line) you can see that the curvature isn't quite the same as a parabola.

2. Now select the second example, showing sinh(x) and its derivative. Do these look familiar? Switch back and forth between the first and second examples. What do you notice? You should see that and that . Notice that, unlike the case with the regular trigonometric sine and cosine, there is no additional minus sign introduced when taking the derivative of cosh.

3. Select the third example, showing tanh(x) and its derivative.You should be able to calculate the derivative from the definition of tanh(x) and the quotient rule. It is .

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© Copyright 2001 David J. Eck

© Copyright 2007 Thomas S. Downey

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Citation: tdowney. (2009, August 06). Hyperbolic Functions. Retrieved December 11, 2013, from Notre Dame OpenCourseWare Web site: http://ocw.nd.edu/mathematics/elements-of-calculus-i/calculus-applets-website/Hyperbolic%20Functions.html.
Copyright 2012, Thomas S. Downey. This work is licensed under a Creative Commons Attribution 3.0 License.