Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)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 7.2.35
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
35. y = ln((x²+1)^5/√(1-x))
Verified step by step guidance1
Rewrite the function to simplify differentiation by expressing the logarithm of a quotient and powers as sums and differences: use the property \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\) and \(\ln(a^n) = n \ln(a)\). So rewrite \(y = \ln\left(\frac{(x^2+1)^5}{\sqrt{1-x}}\right)\) as \(y = 5 \ln(x^2 + 1) - \frac{1}{2} \ln(1 - x)\).
Differentiate each term separately with respect to \(x\) using the chain rule. For the first term, \(5 \ln(x^2 + 1)\), the derivative is \(5 \cdot \frac{1}{x^2 + 1} \cdot \frac{d}{dx}(x^2 + 1)\).
Calculate the derivative inside the chain rule for the first term: \(\frac{d}{dx}(x^2 + 1) = 2x\).
For the second term, \(-\frac{1}{2} \ln(1 - x)\), differentiate using the chain rule: \(-\frac{1}{2} \cdot \frac{1}{1 - x} \cdot \frac{d}{dx}(1 - x)\).
Calculate the derivative inside the chain rule for the second term: \(\frac{d}{dx}(1 - x) = -1\). Then combine all parts to write the derivative \(\frac{dy}{dx}\) as the sum of these differentiated terms.
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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 a function to simplify the differentiation process, especially when the function is a product, quotient, or power. It transforms complicated expressions into sums, differences, and products, making derivatives easier to compute.
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Logarithmic Differentiation
Chain Rule
The chain rule is used to differentiate 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 functions like ln(u(x)) or powers of functions.
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05:02Intro to the Chain Rule
Properties of Logarithms
Properties of logarithms, such as ln(a^b) = b ln(a) and ln(a/b) = ln(a) - ln(b), allow simplification of complex logarithmic expressions. Applying these properties helps break down the given function into simpler terms before differentiation.
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05:36Change of Base Property
Related Practice
Textbook Question
41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.
a. How much longer would it take the soup to cool to 35°C?
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
23. y=arcsin(√2t)
Textbook Question
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
16. y = (ln x)³
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Textbook Question
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
57. lim (x → 0⁺) x^(-1/ln x)