Textbook Question
Evaluate the integrals in Exercises 39–56.
56. ∫sec(x)dx/√(ln(sec(x)+tan(x)))
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 7.3.73
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
Verified step by step guidance1
Recall that the logarithm with base 4 can be rewritten using the change of base formula: \(\log_4 x = \frac{\ln x}{\ln 4}\), where \(\ln\) is the natural logarithm.
Rewrite the given function \(y = \log_4 x + \log_4 x^2\) as \(y = \frac{\ln x}{\ln 4} + \frac{\ln x^2}{\ln 4}\).
Use the logarithm power rule on \(\ln x^2\) to simplify it to \(2 \ln x\), so the function becomes \(y = \frac{\ln x}{\ln 4} + \frac{2 \ln x}{\ln 4}\).
Combine the terms to get \(y = \frac{\ln x + 2 \ln x}{\ln 4} = \frac{3 \ln x}{\ln 4}\).
Differentiate \(y\) with respect to \(x\) using the derivative of \(\ln x\), which is \(\frac{1}{x}\), and treat \(\frac{3}{\ln 4}\) as a constant multiplier. So, \(\frac{dy}{dx} = \frac{3}{\ln 4} \cdot \frac{1}{x}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Functions and Their Properties
Logarithmic functions are the inverses of exponential functions and have specific properties such as log_b(xy) = log_b(x) + log_b(y) and log_b(x^n) = n log_b(x). Understanding these properties helps simplify expressions before differentiation.
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Properties of Functions
Change of Base Formula
The change of base formula, log_b(x) = ln(x) / ln(b), allows rewriting logarithms with any base in terms of natural logarithms, which are easier to differentiate using standard rules.
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Derivative of Logarithmic Functions
The derivative of ln(x) with respect to x is 1/x. Using the chain rule and the change of base formula, the derivative of log_b(x) is 1 / (x ln(b)), which is essential for finding the derivative of logarithms with bases other than e.
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Related Practice
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Textbook Question
80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.
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