Textbook Question
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
8. cosh(3x) - sinh(3x)
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.2.27
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
27. y = θ(sin(lnθ) + cos(lnθ))
Verified step by step guidance1
Identify the function given: \(y = \theta \left( \sin(\ln \theta) + \cos(\ln \theta) \right)\). We need to find \(\frac{dy}{d\theta}\) since the variable is \(\theta\).
Recognize that \(y\) is a product of two functions of \(\theta\): \(u = \theta\) and \(v = \sin(\ln \theta) + \cos(\ln \theta)\). We will use the product rule for differentiation: \(\frac{d}{d\theta}(uv) = u'v + uv'\).
Compute the derivative of \(u = \theta\), which is \(u' = 1\).
Compute the derivative of \(v = \sin(\ln \theta) + \cos(\ln \theta)\). Use the chain rule for each term: For \(\sin(\ln \theta)\), derivative is \(\cos(\ln \theta) \cdot \frac{1}{\theta}\); for \(\cos(\ln \theta)\), derivative is \(-\sin(\ln \theta) \cdot \frac{1}{\theta}\). Then sum these results to find \(v'\).
Apply the product rule: \(\frac{dy}{d\theta} = u'v + uv' = 1 \cdot \left( \sin(\ln \theta) + \cos(\ln \theta) \right) + \theta \cdot \left( \cos(\ln \theta) \cdot \frac{1}{\theta} - \sin(\ln \theta) \cdot \frac{1}{\theta} \right)\). Simplify the expression as much as possible.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chain Rule
The chain rule is used to differentiate composite functions. When a function is nested inside another, such as sin(lnθ), you differentiate the outer function first and multiply by the derivative of the inner function. This rule is essential for handling functions like sin(lnθ) and cos(lnθ).
Recommended video:
05:02
Intro to the Chain Rule
Product Rule
The product rule is applied when differentiating the product of two functions, such as θ multiplied by (sin(lnθ) + cos(lnθ)). It states that the derivative of f(θ)g(θ) is f'(θ)g(θ) + f(θ)g'(θ), allowing you to differentiate each part separately and combine the results.
Recommended video:
05:18The Product Rule
Derivatives of Logarithmic and Trigonometric Functions
Knowing the derivatives of lnθ, sin(x), and cos(x) is crucial. The derivative of lnθ is 1/θ, while the derivatives of sin(x) and cos(x) are cos(x) and -sin(x), respectively. These rules help differentiate the inner and outer functions in the expression.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
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
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
23. y = (x²+1)sech(ln x)
(Hint: Before differentiating, express in terms of exponentials and simplify.)
Textbook Question
77. The region in the first quadrant bounded by the coordinate axes, the line y=3, and the curve x=2/√(y+1) is revolved about the y-axis to generate a solid. Find the volume of the solid.
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.
1. sinh x = -3/4
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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