Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
49. lim (x → 0) (x - sin x) / (x tan x)
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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 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 7.2.61
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
61. y = √(θ + 3) sin θ
Verified step by step guidance1
Start by expressing the function clearly: \(y = \sqrt{\theta + 3} \sin \theta\). Recognize that this is a product of two functions of \(\theta\): \(u(\theta) = \sqrt{\theta + 3}\) and \(v(\theta) = \sin \theta\).
Rewrite \(y\) in a form that is easier to differentiate logarithmically: \(y = (\theta + 3)^{\frac{1}{2}} \sin \theta\).
Take the natural logarithm of both sides to apply logarithmic differentiation: \(\ln y = \ln \left( (\theta + 3)^{\frac{1}{2}} \sin \theta \right)\).
Use logarithm properties to separate the terms: \(\ln y = \frac{1}{2} \ln (\theta + 3) + \ln (\sin \theta)\).
Differentiate both sides with respect to \(\theta\), remembering that \(\frac{d}{d\theta} (\ln y) = \frac{1}{y} \frac{dy}{d\theta}\), and apply the chain rule and derivative rules for logarithms and trigonometric functions.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of functions by taking the natural logarithm of both sides. This simplifies the differentiation process by converting multiplication into addition and powers into products, making it easier to apply derivative rules.
Recommended video:
06:30
Logarithmic Differentiation
Product Rule
The product rule is a fundamental differentiation rule used when finding the derivative of a product of two functions. It states that the derivative of f(θ)g(θ) is f'(θ)g(θ) + f(θ)g'(θ), which helps in differentiating expressions like y = √(θ + 3) sin θ.
Recommended video:
05:18The Product Rule
Chain Rule
The chain rule is used to differentiate composite functions, where one function is inside another. For example, when differentiating √(θ + 3), the chain rule helps by differentiating the outer function (square root) and multiplying by the derivative of the inner function (θ + 3).
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
12. lim (x → ∞) (x - 8x²) / (12x² + 5x)
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
33. y=ln(arctan(x))
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)
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Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
29. y=arcsec(1/t), 0<t<1