James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 3.11 Hyperbolic Functions 263
Therefore
This shows that
y − lnse y d − lnsx 1 sx 2 1 1d
sinh 21 x − lnsx 1 sx 2 1 1d
(See Exercise 25 for another method.)
■
6 Derivatives of Inverse Hyperbolic Functions
Notice that the formulas for the
derivatives of tanh 21 x and coth 21 x
appear to be identical. But the domains
of these functions have no numbers in
common: tanh 21 x is defined for | x | , 1,
whereas coth 21 x is defined for | x | . 1.
d
dx ssinh21 xd −
d
dx scosh21 xd −
1
s1 1 x 2
1
sx 2 2 1
d
1
dx scsch21 xd − 2
| x |sx 2 1 1
d
dx ssech21 xd − 2
d
dx stanh21 xd − 1
d
1 2 x 2 dx scoth2 1
xd − 1
1 2 x 2
1
xs1 2 x 2
The inverse hyperbolic functions are all differentiable because the hyperbolic functions
are differentiable. The formulas in Table 6 can be proved either by the method for
inverse functions or by differentiating Formulas 3, 4, and 5.
ExamplE 4 Prove that d dx ssinh21 xd −
1
s1 1 x 2 .
SOLUTION 1 Let y − sinh 21 x. Then sinh y − x. If we differentiate this equation implicitly
with respect to x, we get
cosh y dy
dx − 1
Since cosh 2 y 2 sinh 2 y − 1 and cosh y > 0, we have cosh y − s1 1 sinh 2 y , so
dy
dx − 1
cosh y − 1
−
s1 1 sinh 2 y
1
s1 1 x 2
SOLUTION 2 From Equation 3 (proved in Example 3), we have
d
dx ssinh21 xd − d dx lnsx 1 sx 2 1 1d
1
−
x 1 sx 2 1 1
−
1
x 1 sx 2 1 1
S1 1
d
dx sx 1 sx 2 1 1d
x
sx 2 1 1D
−
−
sx 2 1 1 1 x
(x 1 sx 2 1 1)sx 2 1 1
1
sx 2 1 1
■
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.