Domanda del libro di testo
Second Derivatives
In Exercises 19–26, use implicit differentiation to find dy/dx and then d²y/dx². Write the solutions in terms of x and y only.
2√y = x – y
219
views
Ch. 3 - Derivatives
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 3, Problema 3.8.3
Assume that y = 5x and dx/dt = 2. Find dy/dt
Guida verificata passo dopo passo1
Start by identifying the given information: y = 5x and dx/dt = 2. We need to find dy/dt.
Recognize that y is a function of x, and both y and x are functions of time t. This implies we need to use the chain rule for differentiation.
Apply the chain rule: The derivative of y with respect to t, dy/dt, can be found using dy/dt = (dy/dx) * (dx/dt).
Differentiate y = 5x with respect to x to find dy/dx. Since y = 5x, dy/dx = 5.
Substitute dy/dx = 5 and dx/dt = 2 into the chain rule formula: dy/dt = 5 * 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
A derivative represents the rate at which a function is changing at any given point and is a fundamental tool in calculus. In this context, dy/dt and dx/dt are derivatives that describe how y and x change with respect to time t. Understanding derivatives is crucial for solving problems involving rates of change.
Video consigliato:
Percorso guidato
05:44
Derivatives
Chain Rule
The chain rule is a formula for computing the derivative of the composition of two or more functions. It is essential here because y is a function of x, which is itself a function of t. The chain rule allows us to find dy/dt by multiplying the derivative of y with respect to x (dy/dx) by the derivative of x with respect to t (dx/dt).
Video consigliato:
05:02Intro to the Chain Rule
Linear Functions
A linear function is a function of the form y = mx + b, where m and b are constants. In this problem, y = 5x is a linear function with a slope of 5. Understanding linear functions helps in recognizing that the derivative dy/dx is constant, which simplifies the application of the chain rule to find dy/dt.
Video consigliato:
Percorso guidato
07:17Linearization
Pratica correlata
Domanda del libro di testo
In Exercises 41–58, find dy/dt.
y = (1 + cos(2t))⁻⁴
257
views
Domanda del libro di testo
In Exercises 41–58, find dy/dt.
y = 4 sin(√(1 + √t))
270
views
Domanda del libro di testo
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together with their tangent lines at the given values of x. Label each curve and tangent line with its equation.
y = sin x, −3π/2 ≤ x ≤ 2π
x = −π, 0, 3π/2
229
views
Domanda del libro di testo
Find the first and second derivatives of the functions in Exercises 33–38.
s = (t² + 5t − 1) / t²
249
views
Domanda del libro di testo
In Exercises 43–50, find by implicit differentiation.
y² = x .
x + 1
261
views