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.
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 7.2.25
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
25. y = ln(ln(x))
Verified step by step guidance1
Identify the function given: \(y = \ln(\ln(x))\). This is a composition of two functions: the natural logarithm of another natural logarithm.
Apply the chain rule for derivatives. The chain rule states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\).
Set the outer function as \(f(u) = \ln(u)\) and the inner function as \(u = \ln(x)\). Then, find the derivative of the outer function with respect to \(u\): \(f'(u) = \frac{1}{u}\).
Find the derivative of the inner function \(u = \ln(x)\) with respect to \(x\): \(u' = \frac{1}{x}\).
Combine these results using the chain rule: \(\frac{dy}{dx} = \frac{1}{\ln(x)} \cdot \frac{1}{x}\). This expresses the derivative of \(y\) with respect to \(x\).
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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 a fundamental technique for finding the derivative of composite functions. It states that the derivative of a function composed of two functions is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
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05:02
Intro to the Chain Rule
Derivative of the Natural Logarithm Function
The derivative of the natural logarithm function ln(x) with respect to x is 1/x, provided x > 0. This rule is essential when differentiating expressions involving logarithms, especially nested logarithms like ln(ln(x)).
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05:18Derivative of the Natural Logarithmic Function
Domain Considerations for Logarithmic Functions
When differentiating logarithmic functions, it's important to consider the domain where the function is defined. For y = ln(ln(x)), both ln(x) and x must be positive, so x > 1. This ensures the function and its derivative are valid and real-valued.
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5:26Graphs of Logarithmic Functions
Related Practice
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)
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
126. eʸ = y^(ln x)
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
7. lim (x → 2) (x - 2) / (x² - 4)
Textbook Question
Solve the differential equation in Exercises 9–22.
11. (dy/dx) = e^(x-y)
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Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
59. y = 2^x