Textbook Question
Evaluate the integrals in Exercises 111–114.
111. ∫₁^(ln x) (1 / t) dt,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 7.2.16
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
16. y = (ln x)³
Verified step by step guidance1
Identify the function given: \(y = (\ln x)^3\). This is a composite function where the outer function is \(u^3\) and the inner function is \(u = \ln x\).
Apply the chain rule for differentiation, which states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).
Differentiate the outer function \(u^3\) with respect to \(u\): \(\frac{d}{du}(u^3) = 3u^2\).
Differentiate the inner function \(u = \ln x\) with respect to \(x\): \(\frac{d}{dx}(\ln x) = \frac{1}{x}\).
Combine the results using the chain rule: \(\frac{dy}{dx} = 3(\ln x)^2 \cdot \frac{1}{x}\).
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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. 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. For example, if y = (ln x)³, treat (ln x) as the inner function and cube as the outer function.
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Intro to the Chain Rule
Derivative of the Natural Logarithm Function
The derivative of ln x with respect to x is 1/x. This is fundamental when differentiating expressions involving logarithms. Knowing this allows you to find the derivative of functions like (ln x)³ by applying the chain rule.
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05:18Derivative of the Natural Logarithmic Function
Power Rule
The power rule states that the derivative of xⁿ is n*xⁿ⁻¹. When differentiating (ln x)³, the power rule applies to the outer function (raising to the third power), which helps simplify the differentiation process when combined with the chain rule.
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Guided course
5:50Power Rules
Related Practice
Textbook Question
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0
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Textbook Question
Evaluate the integrals in Exercises 31–78.
71. ∫(from √2/3 to 2/3)dy/(|y|√(9y²-1))
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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.
2. sinh x = 4/3
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Textbook Question
137. Find a curve through the origin in the xy-plane whose length from x = 0 to x = 1 is L = ∫ from 0 to 1 of sqrt(1 + (1/4)e^x) dx.
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Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = (x^2 - 2x + 2)e^(x)
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