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.
69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))
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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 7.2.14
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
14. y = ln(2θ+2)
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Identify the function given: \(y = \ln(2\theta + 2)\), where \(y\) is expressed in terms of \(\theta\).
Recall the derivative rule for the natural logarithm function: if \(y = \ln(u)\), then \(\frac{dy}{d\theta} = \frac{1}{u} \cdot \frac{du}{d\theta}\).
Set \(u = 2\theta + 2\) and find its derivative with respect to \(\theta\): \(\frac{du}{d\theta} = 2\).
Apply the chain rule by substituting \(u\) and \(\frac{du}{d\theta}\) into the derivative formula: \(\frac{dy}{d\theta} = \frac{1}{2\theta + 2} \cdot 2\).
Simplify the expression if possible to write the derivative in its simplest form.
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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) times the derivative of u. This rule allows us to differentiate logarithmic functions by applying the chain rule to the inner function.
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05:18
Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function 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
Differentiation with Respect to a Variable
When differentiating, it is important to identify the variable with respect to which the derivative is taken. In this problem, the variable is θ, so all derivatives must be computed considering θ as the independent variable.
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07:15Separation of Variables
Related Practice
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
61. y = 5√s
Textbook Question
Evaluate the integrals in Exercises 39–56.
43. ∫(from 0 to π)(sin t)/(2 - cos t) dt
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Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
79. y = θ sin(log₇ θ)
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
31. y=arccot(√t)
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Textbook Question
Let f(x) = x³ − 3x² − 1, x ≥ 2. Find the value of df⁻¹/dx at the point x = −1 = f(3).