Textbook Question
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
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 7.3.122
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Verified step by step guidance1
Recognize that the function is of the form \(y = (\ln x)^{\ln x}\), which is a variable base raised to a variable exponent. This is a perfect candidate for logarithmic differentiation.
Take the natural logarithm of both sides to simplify the expression: \(\ln y = \ln \left( (\ln x)^{\ln x} \right)\).
Use the logarithm power rule to bring down the exponent: \(\ln y = (\ln x) \cdot \ln (\ln x)\).
Differentiate both sides with respect to \(x\). Remember to use implicit differentiation on the left side and the product rule on the right side: \(\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( (\ln x) \cdot \ln (\ln x) \right)\).
Apply the product rule to the right side: \(\frac{d}{dx} (\ln x) \cdot \ln (\ln x) + (\ln x) \cdot \frac{d}{dx} \ln (\ln x)\). Then find each derivative separately using the chain rule, and finally multiply both sides by \(y\) to solve for \(\frac{dy}{dx}\).
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Video duration:
5mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions where both the base and the exponent are variable expressions. By taking the natural logarithm of both sides, the function is transformed into a product or sum, simplifying differentiation using the chain and product rules.
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Logarithmic Differentiation
Properties of Logarithms
Understanding logarithm properties, such as ln(a^b) = b ln(a), is essential for rewriting complex expressions. These properties allow the exponent to be brought down as a multiplier, making differentiation more straightforward when dealing with functions like (ln x)^(ln x).
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05:36Change of Base Property
Chain Rule and Product Rule
The chain rule is used to differentiate composite functions, while the product rule applies when differentiating products of functions. In logarithmic differentiation, after applying logarithms, these rules are often combined to find the derivative of the transformed expression.
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05:18The Product Rule
Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
32. lim (x → 0) (3^x - 1) / (2^x - 1)
1
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Textbook Question
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0
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15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
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Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
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Textbook Question
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt