Textbook Question
Evaluate the integrals in Exercises 31–78.
39. ∫(from 0 to π)tan(x/3)dx
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.PE.7
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
7. y = log₂(x²/2)
Verified step by step guidance1
Recall that the function is given as \(y = \log_2\left(\frac{x^2}{2}\right)\). Our goal is to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).
Use the change of base formula for logarithms to rewrite the function in terms of natural logarithms: \(y = \frac{\ln\left(\frac{x^2}{2}\right)}{\ln(2)}\).
Since \(\ln(2)\) is a constant, the derivative of \(y\) with respect to \(x\) is \(\frac{1}{\ln(2)}\) times the derivative of \(\ln\left(\frac{x^2}{2}\right)\) with respect to \(x\).
Apply the chain rule to differentiate \(\ln\left(\frac{x^2}{2}\right)\). The derivative of \(\ln(u)\) with respect to \(x\) is \(\frac{1}{u} \cdot \frac{du}{dx}\), where \(u = \frac{x^2}{2}\).
Calculate \(\frac{du}{dx}\) where \(u = \frac{x^2}{2}\). Then, combine all parts to express \(\frac{dy}{dx} = \frac{1}{\ln(2)} \cdot \frac{1}{\frac{x^2}{2}} \cdot \frac{du}{dx}\).
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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. Understanding properties like the change of base formula and the rules for logarithms (product, quotient, and power) helps simplify expressions before differentiation.
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Properties of Functions
Derivative of Logarithmic Functions
The derivative of a logarithmic function depends on its base. For log base a, the derivative of log_a(u) with respect to x is (1 / (u ln a)) * du/dx, where u is a function of x. This formula is essential for differentiating log₂(x²/2).
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is used to differentiate composite functions. When differentiating log₂(x²/2), you treat the inside function (x²/2) separately and multiply its derivative by the derivative of the outer logarithmic function.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–78.
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In Exercises 125–128 solve the differential equation.
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Textbook Question
Evaluate the integrals in Exercises 31–78.
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Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
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