Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
∫ (e^(1/x) / x²) dx
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.3.81
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
81. y = log₁₀(e^x)
Guida verificata passo dopo passo1
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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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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Percorso guidato
06:21
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.
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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
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