Domanda del libro di testo
Evaluate and simplify y'.
y = ln w / w⁵
208
views
Ch. 3 - Derivatives
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 3, Problema 3.R.58
Evaluate and simplify y'.
sin x cos(y−1) = 1/2
Guida verificata passo dopo passo1
First, identify the given equation: \( y' \cdot \sin(x) \cdot \cos(y - 1) = \frac{1}{2} \). Here, \( y' \) represents the derivative of \( y \) with respect to \( x \).
To isolate \( y' \), divide both sides of the equation by \( \sin(x) \cdot \cos(y - 1) \). This gives: \( y' = \frac{1}{2} \div (\sin(x) \cdot \cos(y - 1)) \).
Simplify the expression for \( y' \) by rewriting the division as a multiplication by the reciprocal: \( y' = \frac{1}{2} \cdot \frac{1}{\sin(x) \cdot \cos(y - 1)} \).
Combine the fractions to express \( y' \) as a single fraction: \( y' = \frac{1}{2 \cdot \sin(x) \cdot \cos(y - 1)} \).
The expression for \( y' \) is now simplified. Ensure that the domain restrictions for \( \sin(x) \) and \( \cos(y - 1) \) are considered, as they cannot be zero.
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:
4mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Implicit Differentiation
Implicit differentiation is a technique used to differentiate equations where the dependent variable is not isolated on one side. In this case, we have a relationship between x and y given by sin x cos(y−1) = 1/2. By differentiating both sides with respect to x, we can find dy/dx, which represents y'.
Video consigliato:
Percorso guidato
05:14
Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus that allows us to differentiate composite functions. When differentiating terms involving y, such as cos(y−1), we apply the chain rule to account for the derivative of the inner function (y−1) with respect to x, which involves dy/dx. This is crucial for correctly finding y'.
Video consigliato:
05:02Intro to the Chain Rule
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this problem, understanding the properties of sine and cosine is essential, especially since we are working with sin x and cos(y−1). These identities can help simplify the expression after differentiation and aid in solving for y'.
Video consigliato:
7:17Verifying Trig Equations as Identities
Pratica correlata
Domanda del libro di testo
9–61. Evaluate and simplify y'.
y = tan (sin θ)
266
views
Domanda del libro di testo
9–61. Evaluate and simplify y'.
y = e^6x sin x
295
views
Domanda del libro di testo
5-8. Use differentiation to verify each equation.
d/dx (tan³ x-3 tan x+3x) = 3 tan⁴x
254
views
Domanda del libro di testo
The population of the United States (in millions) by decade is given in the table, where t is the number of years after 1910. These data are plotted and fitted with a smooth curve y = p(t) in the figure. <IMAGE><IMAGE>
Compute the average rate of population growth from 1950 to 1960.
225
views
Domanda del libro di testo
Use differentiation to verify each equation.
d/dx (x⁴ − ln(x⁴ + 1))=4x⁷ / (1 + x⁴).
248
views