Textbook Question
112. True, or false? Give reasons for your answers.
a. 1/x⁴ = O(1/x² + 1/x⁴)
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.P.25
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)
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Start by rewriting the function to make logarithmic differentiation easier. Given \( y = \frac{2(x^2 + 1)}{\sqrt{\cos 2x}} \), express it as \( y = 2(x^2 + 1)(\cos 2x)^{-1/2} \).
Take the natural logarithm of both sides: \( \ln y = \ln 2 + \ln (x^2 + 1) + \ln (\cos 2x)^{-1/2} \).
Simplify the logarithmic expression using log properties: \( \ln y = \ln 2 + \ln (x^2 + 1) - \frac{1}{2} \ln (\cos 2x) \).
Differentiate both sides with respect to \( x \). Remember to use the chain rule on \( \ln y \), which gives \( \frac{1}{y} \frac{dy}{dx} \), and apply the derivatives of each term on the right side.
Solve for \( \frac{dy}{dx} \) by multiplying both sides by \( y \), and then substitute back the original expression for \( y \) to express the derivative in terms of \( x \).
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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 complicated expressions. By taking the natural logarithm of both sides, the differentiation process simplifies using properties of logarithms, such as turning products into sums and powers into multipliers.
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Logarithmic Differentiation
Properties of Logarithms
Understanding the properties of logarithms is essential for logarithmic differentiation. Key properties include log(ab) = log a + log b, log(a/b) = log a - log b, and log(a^n) = n log a. These allow complex expressions to be broken down into simpler parts that are easier to differentiate.
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05:36Change of Base Property
Chain Rule
The chain rule is a fundamental differentiation rule used when differentiating composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This is crucial when differentiating expressions like cos(2x) inside the logarithm.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Textbook Question
112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)
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Textbook Question
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
27. y = (((t+1)(t-1))/((t-2)(t+3)))^5, t>2
Textbook Question
111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))