Derivatives of Hyperbolic Functions
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
.
This work by Thomas S. Downey is licensed under a Creative Commons Attribution 3.0 License.