Textbook Question
Evaluate the integrals in Exercises 97–110.
99. ∫₀³ (√2 + 1)x^(√2) dx
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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 7.7.17
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
17. y = ln(sinh z)
Verified step by step guidance1
Identify the function given: \(y = \ln(\sinh z)\), where \(\sinh z\) is the hyperbolic sine function.
Recall the chain rule for differentiation: if \(y = \ln(u)\), then \(\frac{dy}{dz} = \frac{1}{u} \cdot \frac{du}{dz}\).
Set \(u = \sinh z\). Then find the derivative of \(u\) with respect to \(z\): \(\frac{du}{dz} = \cosh z\), since the derivative of \(\sinh z\) is \(\cosh z\).
Apply the chain rule: \(\frac{dy}{dz} = \frac{1}{\sinh z} \cdot \cosh z\).
Express the derivative as \(\frac{dy}{dz} = \frac{\cosh z}{\sinh z}\), which can also be written as \(\coth z\) if desired.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of the Natural Logarithm Function
The derivative of ln(u), where u is a differentiable function of a variable, is given by (1/u) * (du/dx). This rule allows us to differentiate logarithmic functions by first identifying the inner function and then applying the chain rule.
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Derivative of the Natural Logarithmic Function
Hyperbolic Sine Function and Its Derivative
The hyperbolic sine function, sinh(z), is defined as (e^z - e^(-z))/2. Its derivative with respect to z is cosh(z), which is (e^z + e^(-z))/2. Knowing this derivative is essential when differentiating expressions involving sinh.
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03:53Derivatives of Sine & Cosine
Chain Rule in Differentiation
The chain rule is used to differentiate composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). This rule is crucial when differentiating ln(sinh z), as ln and sinh are nested functions.
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
Find the values in Exercises 9–12.
9. sin(arccos((√2)/2))
Textbook Question
Evaluate the integrals in Exercises 41–60.
59. ∫(from -ln2 to 0)cosh²(x/2) dx
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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
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
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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