Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) 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 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 7.3.81
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)
Verified step by step guidance1
Recognize that the function is given as \(y = \log_{10}(e^x)\), which is a logarithm with base 10 of the expression \(e^x\).
Use the logarithm property that allows you to rewrite \(\log_{10}(e^x)\) as \(x \cdot \log_{10}(e)\), since \(\log_b(a^c) = c \cdot \log_b(a)\).
Recall that \(\log_{10}(e)\) is a constant because it does not depend on \(x\).
Differentiate \(y = x \cdot \log_{10}(e)\) with respect to \(x\) using the constant multiple rule, which states that the derivative of \(c \cdot f(x)\) is \(c \cdot f'(x)\) where \(c\) is a constant.
Since the derivative of \(x\) with respect to \(x\) is 1, the derivative \(\frac{dy}{dx}\) simplifies to \(\log_{10}(e)\).
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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 logₐ(b^x) = x·logₐ(b) helps simplify expressions before differentiation. In this problem, recognizing that log₁₀(e^x) can be rewritten using these properties is essential.
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Properties of Functions
Derivative of Exponential Functions
The derivative of an exponential function e^x with respect to x is e^x. This fundamental rule is crucial when differentiating expressions involving exponentials, especially when combined with logarithmic functions.
Recommended video:
04:50Derivatives of General Exponential Functions
Chain Rule for Differentiation
The chain rule is used to differentiate composite functions. When a function is nested inside another, like log₁₀(e^x), the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
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05:02Intro to the Chain Rule
Related Practice
Textbook Question
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
33. y = csch⁻¹(1/2)^θ
Textbook Question
Evaluate the integrals in Exercises 97–110.
97. ∫ 3x^(√3) dx
Textbook Question
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
3. lim (x → ∞) (5x² - 3x) / (7x² + 1)
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Textbook Question
Evaluate the integrals in Exercises 41–60.
43. ∫6cosh(x/2 - ln3)dx
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
77. y = log₃(((x + 1)/(x − 1))^(ln 3))