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:

- 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.

- 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.

- 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 April 25, 2014, 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.