Textbook Question
Evaluate the integrals in Exercises 53–76.
59. ∫(from 0 to 1)4ds/√(4-s²)
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.2.59
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
59. y = √(t/(t+1))
Verified step by step guidance1
Start by expressing the function in a form that is easier to differentiate using logarithmic differentiation. Given \( y = \sqrt{\frac{t}{t+1}} \), rewrite it as \( y = \left( \frac{t}{t+1} \right)^{\frac{1}{2}} \).
Take the natural logarithm of both sides to apply logarithmic differentiation: \( \ln y = \ln \left( \left( \frac{t}{t+1} \right)^{\frac{1}{2}} \right) \).
Use the logarithm power rule to bring down the exponent: \( \ln y = \frac{1}{2} \ln \left( \frac{t}{t+1} \right) \).
Apply the logarithm quotient rule to separate the terms inside the logarithm: \( \ln y = \frac{1}{2} ( \ln t - \ln (t+1) ) \).
Differentiate both sides with respect to \( t \). Remember that \( \frac{d}{dt} (\ln y) = \frac{1}{y} \frac{dy}{dt} \) by the chain rule. Then differentiate the right side term-by-term using the derivative of \( \ln t \) and \( \ln (t+1) \).
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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 taking the natural logarithm of both sides of an equation y = f(x) to simplify the differentiation process, especially for functions involving products, quotients, or powers. After applying the log, implicit differentiation is used to find dy/dx, making complex derivatives easier to handle.
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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. This is essential when differentiating expressions like square roots or powers.
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05:02Intro to the Chain Rule
Quotient Rule and Simplification
The quotient rule is used to differentiate functions expressed as a ratio of two differentiable functions. It states that the derivative of f(x)/g(x) is (g(x)f'(x) - f(x)g'(x)) / [g(x)]². In logarithmic differentiation, recognizing the quotient inside the logarithm helps simplify the derivative by converting division into subtraction.
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06:43The Quotient Rule
Related Practice
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