Domanda del libro di testo
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
53. lim (x → 1⁺) x^(1/(1 - x))
27
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
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 7.2.23
In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
23. y = ln(x)/(1+ln(x))
Guida verificata passo dopo passo1
Identify the function given: \(y = \frac{\ln(x)}{1 + \ln(x)}\). We need to find \(\frac{dy}{dx}\) since the variable is \(x\).
Recognize that this is a quotient of two functions: numerator \(u = \ln(x)\) and denominator \(v = 1 + \ln(x)\).
Recall the quotient rule for derivatives: \(\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2}\).
Compute the derivatives of numerator and denominator separately: \(u' = \frac{d}{dx} \ln(x) = \frac{1}{x}\) and \(v' = \frac{d}{dx} (1 + \ln(x)) = \frac{1}{x}\).
Substitute \(u\), \(v\), \(u'\), and \(v'\) into the quotient rule formula to express \(\frac{dy}{dx}\) as \(\frac{(1 + \ln(x)) \cdot \frac{1}{x} - \ln(x) \cdot \frac{1}{x}}{(1 + \ln(x))^2}\).
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Derivative of the Natural Logarithm Function
The derivative of ln(x) with respect to x is 1/x. This fundamental rule is essential when differentiating expressions involving natural logarithms, as it allows us to handle the logarithmic part of the function accurately.
Video consigliato:
05:18
Derivative of the Natural Logarithmic Function
Quotient Rule
The quotient rule is used to differentiate functions expressed as one function divided by another. It states that the derivative of f(x)/g(x) is (g(x)f'(x) - f(x)g'(x)) / [g(x)]², which is crucial for differentiating y = ln(x) / (1 + ln(x)).
Video consigliato:
06:43The Quotient Rule
Chain Rule
The chain rule helps differentiate composite functions by multiplying the derivative of the outer function by the derivative of the inner function. It is important here because ln(x) appears inside another function, requiring careful application of this rule.
Video consigliato:
05:02Intro to the Chain Rule
Pratica correlata
Domanda del libro di testo
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
43. y=√(arcsin x)
29
views
Domanda del libro di testo
"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"
25
views
Domanda del libro di testo
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
45. y=cos(x-arccos(x))
25
views
Domanda del libro di testo
Use the results of Exercise 55 to show that the functions in Exercises 56–60 have inverses over their domains. Find a formula for df⁻¹/dx using Theorem 1.
f(x) = 27x³
18
views
Domanda del libro di testo
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
44. lim (x → 0⁺) (csc x - cot x + cos x)
39
views