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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.2.59
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))

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1
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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Intro 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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Related Practice
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