The last set of functions that we’re going to be looking in
this chapter at are the hyperbolic functions. In many physical situations combinations of and arise fairly often. Because of this these combinations are given
names. There are the six hyperbolic functions and they are defined as follows.
Here are the graphs of the three main hyperbolic functions.
We also have the following facts about the hyperbolic functions.
You’ll note that these are similar, but not quite the same,
to some of the more common trig identities so be careful to not confuse the
identities here with those of the regular trig functions.
Because the hyperbolic functions are defined in terms
of exponential functions finding their derivatives is fairly simple provided
you’ve already read through the next section.
We haven’t however so we’ll need the following formula that can be
easily proved after we’ve covered the next section.
With this formula we’ll do the derivative for hyperbolic
sine and leave the rest to you as an exercise.
For the rest we can either use the definition of the
hyperbolic function and/or the
quotient rule. Here are all six
derivatives.
Here are a couple of quick derivatives using hyperbolic functions.
Example 1 Differentiate
each of the following functions.
(a)
(b)
Solution
(a)
(b)
