Textbook Question
Find the areas between the curves y=2(log_2(x))/x and y=2(log_4(x))/x and the x-axis from x=1 to x=e. What is the ratio of the larger area to the smaller?
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.P.13
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Verified step by step guidance1
Recognize that the function is of the form \(y = f(x)^{g(x)}\), where both the base and the exponent depend on \(x\). This suggests using logarithmic differentiation.
Take the natural logarithm of both sides: \(\ln y = \ln \left( (x+2)^{x+2} \right)\).
Use the logarithm power rule to simplify the right side: \(\ln y = (x+2) \cdot \ln (x+2)\).
Differentiate both sides with respect to \(x\). For the left side, use implicit differentiation: \(\frac{1}{y} \frac{dy}{dx}\). For the right side, apply the product rule to \((x+2) \cdot \ln (x+2)\).
After differentiating, solve for \(\frac{dy}{dx}\) by multiplying both sides by \(y\), and then substitute back \(y = (x+2)^{x+2}\) to express the derivative in terms of \(x\).
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3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Implicit Differentiation and Logarithmic Differentiation
When a function has a variable in both the base and the exponent, such as y = (x+2)^(x+2), logarithmic differentiation is used. Taking the natural logarithm of both sides simplifies the expression, allowing differentiation of the exponent and base separately.
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06:30
Logarithmic Differentiation
Derivative of Exponential Functions with Variable Exponents
For functions where the exponent is a variable, the derivative involves applying the chain rule and product rule after rewriting the function using logarithms. This approach helps handle the complexity of differentiating expressions like a(x)^{b(x)}.
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04:50Derivatives of General Exponential Functions
Chain Rule and Product Rule
The chain rule is used to differentiate composite functions, while the product rule applies when differentiating products of functions. Both are essential here because the function involves a product of terms after logarithmic transformation.
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05:18The Product Rule
Related Practice
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ
Textbook Question
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)
Textbook Question
111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))
Textbook Question
7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?