Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.7.2
Each of Exercises 1–4 gives a value of sinh x or cosh x. Use the definitions and the identity cosh²x - sinh²x = 1 to find the values of the remaining five hyperbolic functions.
2. sinh x = 4/3
Verified step by step guidance1
Recall the fundamental identity for hyperbolic functions: \(\cosh^{2}x - \sinh^{2}x = 1\). Given \(\sinh x = \frac{4}{3}\), substitute this value into the identity to find \(\cosh x\).
Calculate \(\cosh x\) by rearranging the identity: \(\cosh^{2}x = 1 + \sinh^{2}x\). Substitute \(\sinh x = \frac{4}{3}\) to get \(\cosh^{2}x = 1 + \left(\frac{4}{3}\right)^{2}\).
Take the positive square root of \(\cosh^{2}x\) to find \(\cosh x\), since \(\cosh x\) is always positive for real \(x\).
Use the definitions of the other hyperbolic functions in terms of \(\sinh x\) and \(\cosh x\) to find their values: \(\tanh x = \frac{\sinh x}{\cosh x}\), \(\coth x = \frac{\cosh x}{\sinh x}\), \(\sech x = \frac{1}{\cosh x}\), and \(\csch x = \frac{1}{\sinh x}\).
Substitute the known values of \(\sinh x\) and \(\cosh x\) into these formulas to express all five remaining hyperbolic functions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions Definitions
Hyperbolic functions such as sinh x and cosh x are defined using exponential functions: sinh x = (e^x - e^(-x))/2 and cosh x = (e^x + e^(-x))/2. Understanding these definitions helps in expressing and manipulating hyperbolic functions algebraically.
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Definition of the Definite Integral
Fundamental Hyperbolic Identity
The identity cosh²x - sinh²x = 1 is analogous to the Pythagorean identity in trigonometry. It allows you to find one hyperbolic function if the other is known, which is essential for solving problems involving multiple hyperbolic functions.
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Other Hyperbolic Functions and Their Relationships
Besides sinh and cosh, the other hyperbolic functions are tanh x, coth x, sech x, and csch x, defined as ratios involving sinh and cosh. Knowing how to express these functions in terms of sinh and cosh is crucial for finding their values once sinh x or cosh x is given.
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Related Practice
Textbook Question
82. Find a curve through the point (1, 0) whose length from x=1 to x=2 is
L = ∫(from 1 to 2)√(1 + 1/x²)dx.
Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
75. lim (x → ∞) e^(x²) / (x e^x)
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Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
115. y = (x + 1)ˣ
Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
13. y = 6sinh(x/3)
Textbook Question
Verify the integration formulas in Exercises 37–40.
39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C
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