Textbook Question
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
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 7.2.23
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
23. y = ln(x)/(1+ln(x))
Verified step by step guidance1
Identify the function given: \(y = \frac{\ln(x)}{1 + \ln(x)}\). We need to find \(\frac{dy}{dx}\) since the variable is \(x\).
Recognize that this is a quotient of two functions: numerator \(u = \ln(x)\) and denominator \(v = 1 + \ln(x)\).
Recall the quotient rule for derivatives: \(\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}\).
Compute the derivatives of numerator and denominator separately: \(u' = \frac{d}{dx} \ln(x) = \frac{1}{x}\) and \(v' = \frac{d}{dx} (1 + \ln(x)) = \frac{1}{x}\).
Substitute \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula to express \(\frac{dy}{dx}\) as \(\frac{(1 + \ln(x)) \cdot \frac{1}{x} - \ln(x) \cdot \frac{1}{x}}{(1 + \ln(x))^2}\).
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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(x) with respect to x is 1/x. This fundamental rule is essential when differentiating expressions involving natural logarithms, as it allows us to handle the logarithmic part of the function accurately.
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Derivative of the Natural Logarithmic Function
Quotient Rule
The quotient rule is used to differentiate functions expressed as one function divided by another. It states that the derivative of f(x)/g(x) is (g(x)f'(x) - f(x)g'(x)) / [g(x)]², which is crucial for differentiating y = ln(x) / (1 + ln(x)).
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06:43The Quotient Rule
Chain Rule
The chain rule helps differentiate composite functions by multiplying the derivative of the outer function by the derivative of the inner function. It is important here because ln(x) appears inside another function, requiring careful application of this rule.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
Evaluate the integrals in Exercises 87–96.
89. ∫₀¹ 2^(−θ) dθ
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
43. y=√(arcsin x)
Textbook Question
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
45. y=cos(x-arccos(x))
Textbook Question
Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.
f(x) = 27x³