Textbook Question
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
21. y = ln(cosh v) - 1/2 tanh²v
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 7.3.65
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
65. y = (cos θ)^(√2)
Verified step by step guidance1
Identify the function given: \(y = (\cos \theta)^{\sqrt{2}}\). Here, \(y\) is a function of \(\theta\).
Recognize that the function is of the form \(y = [u(\theta)]^n\) where \(u(\theta) = \cos \theta\) and \(n = \sqrt{2}\), a constant exponent.
Apply the general power rule combined with the chain rule for differentiation: \(\frac{dy}{d\theta} = n [u(\theta)]^{n-1} \cdot \frac{du}{d\theta}\).
Calculate the derivative of the inner function: \(\frac{d}{d\theta}(\cos \theta) = -\sin \theta\).
Substitute back into the formula to express the derivative: \(\frac{dy}{d\theta} = \sqrt{2} (\cos \theta)^{\sqrt{2} - 1} (-\sin \theta)\).
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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, where one function is inside another. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). In this problem, since y = (cos θ)^(√2) is a composition of the cosine function raised to a power, the chain rule helps differentiate the outer power function and the inner cosine function.
Recommended video:
05:02
Intro to the Chain Rule
Differentiation of Trigonometric Functions
Understanding how to differentiate trigonometric functions like cosine is essential. The derivative of cos θ with respect to θ is -sin θ. This derivative is a key part of applying the chain rule when differentiating expressions involving trigonometric functions.
Recommended video:
6:04Introduction to Trigonometric Functions
Logarithmic Differentiation
Logarithmic differentiation is useful when differentiating functions with variable exponents, such as (cos θ)^(√2). By taking the natural logarithm of both sides, the exponent can be brought down, simplifying the differentiation process. This technique helps handle powers that are constants or functions.
Recommended video:
06:30Logarithmic Differentiation
Related Practice
Textbook Question
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
6. sinh(2ln x)
Textbook Question
82. For what values of a and b is
lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?
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Textbook Question
44. Silver cooling in air The temperature of an ingot of silver is 60°C above room temperature right now. Twenty minutes ago, it was 70°C above room temperature. How far above room temperature will the silver be
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Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
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Textbook Question
Solve the differential equation in Exercises 9–22.
17. (dy/dx) = 2x(y - 1), y > 1
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