Chapter 2: Differentiation
Section 2.10: The Inverse Hyperbolic Functions and Their Derivatives
Essentials
Introduction
Table 2.10.1 contains a graph of the principal branch for each of the six hyperbolic functions, and the graph of the corresponding inverse function.
Principal Branch
Principal Domain
Inverse Function
−∞<x<∞
0≤x<∞
−∞,0⋃0,∞
Table 2.10.1 Principal branches of the hyperbolic functions, and their inverse functions
Table 2.10.2 lists the derivatives of the six inverse hyperbolic functions.
Function
Maple's Derivative
Textbook Derivative
sinh−1x
ⅆⅆ x arcsinhx = 11+x2
11+x2
cosh−1x
ⅆⅆ x arccoshx = 1x−1x+1
1x2−1, x>1
tanh−1x
ⅆⅆ x arctanhx = 11−x2
11−x2, x<1
coth−1x
ⅆⅆ x arccothx = −1x2−1
11−x2, x>1
sech−1x
ⅆⅆ x arcsechx = −1x21x−11x+1
−1x 1− x2, x<1
csch−1x
ⅆⅆ x arccschx = −1x21+1x2
−1x 1+x2
Table 2.10.2 Derivatives of the six inverse hyperbolic functions
Maple's differentiation formulas are correct for x complex, but in the typical calculus text, the formulas are stated for x real. That is why the textbook formulas have restrictions and vary slightly from the Maple form.
The second column in the table uses the formal name of each inverse function. However, in 2D math mode, Maple understands the usage sinh−1x for arcsinhx, etc.
Graphs of the Inverse Hyperbolic Functions and their Derivatives
Select Functionarcsinh(x)arccosh(x)arctanh(x)arccoth(x)arcsech(x)arccsch(x)
Figure 2.10.1 Graph of inverse hyperbolic function (red) and its derivative (blue)
Figure 2.10.1 can contain a graph of one of the inverse hyperbolic functions (drawn in red) and its derivative (drawn in blue). Simply select a function from the drop-down box below the space where the graph is to appear.
Alternatively, select an inverse hyperbolic function from the drop-down box in Table 2.10.3, then press the button "Launch Tutor" to launch the Derivatives tutor with the corresponding inverse function embedded.
The graphs for arccothx and arccschx are slightly more refined in Figure 2.10.1.
arcsinh(x)arccosh(x)arctanh(x)arccoth(x)arcsech(x)arccsch(x)
Table 2.10.3 Access the Derivatives tutor
Logarithmic Form of the Inverse Hyperbolic Functions
Table 2.10.4 lists the logarithmic form for each of the inverse hyperbolic functions.
arcsinhx=lnx+x2+1
arccoshx=lnx+x−1x+1
arctanhx=12lnx+1−12ln1−x
arccothx=12lnx+1−12lnx−1
arcsechx=ln1x+1x−11x+1
arccschx=ln1x+1+1x2
Table 2.10.4 Logarithmic form of inverse hyperbolic functions
See, for example, Example 2.10.1.
Précis
The graphs of the principal branches of the hyperbolic functions and the graphs of the corresponding inverse functions are given in Table 2.10.1.
The derivatives of the inverse hyperbolic functions are listed in Table 2.10.2.
Examples
Example 2.10.1
Establish the identity arcsinhx=lnx+x2+1.
Example 2.10.2
Evaluate ⅆⅆx tanh−1x.
Example 2.10.3
Evaluate ⅆⅆx sinh−1x2.
Example 2.10.4
Evaluate ⅆⅆx sec−11/x.
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