Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
40
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 7.3.71
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
71. y = log₂(5θ)
Guida verificata passo dopo passo1
Recall the formula for the derivative of a logarithm with an arbitrary base: if \( y = \log_a(u) \), then \( \frac{dy}{dx} = \frac{1}{u \ln(a)} \cdot \frac{du}{dx} \), where \( \ln \) is the natural logarithm.
Identify the base \( a \) and the argument \( u \) in the given function. Here, \( y = \log_2(5\theta) \), so \( a = 2 \) and \( u = 5\theta \).
Compute the derivative of the inside function \( u = 5\theta \) with respect to \( \theta \). Since \( 5 \) is a constant, \( \frac{du}{d\theta} = 5 \).
Apply the derivative formula: \( \frac{dy}{d\theta} = \frac{1}{5\theta \ln(2)} \times 5 \).
Simplify the expression by canceling common factors if possible to express the derivative in its simplest form.
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 Logarithmic Functions
The derivative of a logarithmic function depends on its base. For a logarithm with base a, log_a(x), the derivative is 1/(x ln(a)) times the derivative of x. Understanding this allows you to differentiate log₂(5θ) by applying the chain rule.
Video consigliato:
05:18
Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is used to differentiate composite functions. When a function is inside another, like 5θ inside the logarithm, you first differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function.
Video consigliato:
05:02Intro to the Chain Rule
Properties of Logarithms
Logarithmic properties, such as log_a(bx) = log_a(b) + log_a(x), can simplify differentiation. Recognizing these properties helps break down complex logarithmic expressions into simpler parts, making differentiation more straightforward.
Video consigliato:
05:36Change of Base Property
Pratica correlata
Domanda del libro di testo
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))
27
views
Domanda del libro di testo
Evaluate the integrals in Exercises 53–76.
69. ∫dx/((2x-1)√((2x-1)²-4))
27
views
Domanda del libro di testo
Evaluate the integrals in Exercises 33–54.
49. ∫ e^(sec πt) sec πt tan πt dt
23
views
Domanda del libro di testo
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
47. y=(arccot(x³))³
29
views
Domanda del libro di testo
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)
32
views