The three hyperbolic functions are defined as:

The applet below shows the graphs of these functions and their derivatives.

See About the calculus applets for operating instructions. |

In the above applet, there is a pull-down menu at the top to select which function you would like to explore. The selected function is plotted in the left window and its derivative on the right.

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.

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

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:

- Constant, Line, and Power Functions
- Exponential Functions
- Trigonometric Functions
- Constant Multiple
- Combinations: Sum and Difference
- Combinations: Product and Quotient
- Composition of Functions (the Chain Rule)
- Transformations of Functions
- Inverses of Functions
- Hyperbolic Functions
- Linear Approximation
- Mean Value Theorem

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