3
Differentiation Rules
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3.11
Hyperbolic Functions
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Hyperbolic Functions and Their Derivatives
3
Hyperbolic Functions and Their Derivatives (1 of 5)
Certain combinations of the exponential functions
arise so frequently in mathematics and its applications that they deserve to be given special names.
In many ways they are analogous to the trigonometric functions, and they have the same relationship to the hyperbola that the trigonometric functions have to the circle.
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Hyperbolic Functions and Their Derivatives (2 of 5)
For this reason they are collectively called hyperbolic functions and individually called hyperbolic sine, hyperbolic cosine, and so on.
Definition of the Hyperbolic Functions
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Hyperbolic Functions and Their Derivatives (3 of 5)
The hyperbolic functions satisfy a number of identities that are similar to well-known trigonometric identities.
We list some of them here and leave most of the proofs to the exercises.
Hyperbolic Identities
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Example 1
Prove (a)
(b)
Solution:
(a)
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Example 1 – Solution
(b) We start with the identity proved in part (a):
If we divide both sides by
we get
or
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Hyperbolic Functions and Their Derivatives (4 of 5)
The derivatives of the hyperbolic functions are easily computed. For example,
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Hyperbolic Functions and Their Derivatives (5 of 5)
We list the differentiation formulas for the hyperbolic functions as Table 1.
1 Derivatives of Hyperbolic Functions
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Example 2
If
Solution: Using (1) and the Chain Rule, we have
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Inverse Hyperbolic Functions and Their Derivatives
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Inverse Hyperbolic Functions and Their Derivatives (1 of 5)
The sin h and tanh are one-to-one functions and so they have inverse functions
denoted by
The cos h is not one-to-one, but when restricted
to the domain
it becomes one-to-one.
The inverse hyperbolic cosine function is defined as the inverse of this restricted function.
2
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Inverse Hyperbolic Functions and Their Derivatives (2 of 5)
We can sketch the graphs of
in Figures 8, 9, and 10.
Figure 8
Figure 9
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Inverse Hyperbolic Functions and Their Derivatives (3 of 5)
Figure 10
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Inverse Hyperbolic Functions and Their Derivatives (4 of 5)
Since the hyperbolic functions are defined in terms of exponential functions, it’s not surprising to learn that the inverse hyperbolic functions can be expressed in terms of logarithms.
In particular, we have:
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Example 3
Show that
Solution:
Let
so
or, multiplying by
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Example 3 – Solution (1 of 2)
This is really a quadratic equation in
Solving by the quadratic formula, we get
Note that
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Example 3 – Solution (2 of 2)
Thus the minus sign is inadmissible and we have
Therefore
This shows that
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Inverse Hyperbolic Functions and Their Derivatives (5 of 5)
6 Derivatives of Inverse Hyperbolic Functions
The inverse hyperbolic functions are all differentiable because the hyperbolic functions are differentiable.
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Example 4
Prove that
Solution:
Let
Then sin h y = x. If we differentiate this equation implicitly
with respect to x, we get
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Example 4 – Solution
Since
and cos h y ≥ 0, we have
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