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.
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 7.7.13
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
13. y = 6sinh(x/3)
Verified step by step guidance1
Recall the definition of the hyperbolic sine function: \(\sinh(u)\), where \(u\) is a function of \(x\). The derivative of \(\sinh(u)\) with respect to \(x\) is \(\cosh(u) \cdot \frac{du}{dx}\).
Identify the inner function \(u = \frac{x}{3}\). We will need to find \(\frac{du}{dx}\).
Calculate the derivative of the inner function: \(\frac{du}{dx} = \frac{d}{dx} \left( \frac{x}{3} \right) = \frac{1}{3}\).
Apply the chain rule to differentiate \(y = 6 \sinh \left( \frac{x}{3} \right)\): \(\frac{dy}{dx} = 6 \cdot \cosh \left( \frac{x}{3} \right) \cdot \frac{1}{3}\).
Simplify the expression for the derivative: \(\frac{dy}{dx} = 2 \cosh \left( \frac{x}{3} \right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Sine Function (sinh)
The hyperbolic sine function, sinh(x), is defined as (e^x - e^(-x))/2. It is similar to the sine function but relates to hyperbolic geometry. Understanding its definition helps in differentiating expressions involving sinh.
Recommended video:
5:53
Graph of Sine and Cosine Function
Chain Rule
The chain rule is a differentiation technique used when a function is composed of another function, such as y = f(g(x)). It states that the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Recommended video:
05:02Intro to the Chain Rule
Derivative of Hyperbolic Functions
The derivative of sinh(x) with respect to x is cosh(x), the hyperbolic cosine function. Knowing this derivative is essential for differentiating expressions involving sinh, especially when combined with the chain rule.
Recommended video:
5:50Asymptotes of Hyperbolas
Related Practice
Textbook Question
In Exercises 5–8, show that each function is a solution of the given initial value problem.
7. Differential Equation: xy' + y = -sin(x), x>0
Initial condition: y(π/2) = 0
Solution candidate: y = cos(x)/x
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
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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