Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x² − 2x, x ≤ 1
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 7.2.29
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
29. y = ln(1/(x√(x+1)))
Verified step by step guidance1
Rewrite the function to simplify differentiation. Given \( y = \ln\left( \frac{1}{x\sqrt{x+1}} \right) \), express it as \( y = \ln(1) - \ln\left(x\sqrt{x+1}\right) \). Since \( \ln(1) = 0 \), this simplifies to \( y = -\ln\left(x\sqrt{x+1}\right) \).
Use the property of logarithms to separate the terms inside the logarithm: \( \ln\left(x\sqrt{x+1}\right) = \ln(x) + \ln\left( (x+1)^{1/2} \right) = \ln(x) + \frac{1}{2} \ln(x+1) \). So, \( y = -\left( \ln(x) + \frac{1}{2} \ln(x+1) \right) \).
Differentiate \( y \) with respect to \( x \) by applying the derivative of logarithmic functions: \( \frac{d}{dx} \ln(x) = \frac{1}{x} \) and \( \frac{d}{dx} \ln(x+1) = \frac{1}{x+1} \).
Apply the chain rule and linearity of differentiation: \( \frac{dy}{dx} = -\left( \frac{1}{x} + \frac{1}{2} \cdot \frac{1}{x+1} \right) \).
Combine the terms to write the derivative as a single expression, if desired, by finding a common denominator or leaving it as a sum of fractions.
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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 involves applying the properties of logarithms to simplify the differentiation of complex functions, especially those involving products, quotients, or powers. It allows rewriting the function as a sum or difference of simpler logarithmic terms, making the derivative easier to find.
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Logarithmic Differentiation
Chain Rule
The chain rule is a fundamental differentiation technique used when dealing with 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.
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05:02Intro to the Chain Rule
Derivative of Logarithmic Functions
The derivative of the natural logarithm function ln(u) with respect to x is (1/u) times the derivative of u. This rule is essential when differentiating functions expressed as logarithms, especially after simplifying the original function using logarithmic properties.
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05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
For problems 49–52 use implicit differentiation to find dy/dx at the given point P.
49. 3arctan(x) + arcsin(y) = π/4; P(1, -1)
1
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Textbook Question
Evaluate the integrals in Exercises 39–56.
45. ∫(from 1 to 2)(2ln x)/x dx
1
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Textbook Question
Evaluate the integrals in Exercises 41–60.
45. ∫tanh(x/7)dx
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x²+1, x≥0
Textbook Question
Which of the functions graphed in Exercises 1–6 are one-to-one, and which are not?