Domanda del libro di testo
Find the derivatives of the functions in Exercises 19–40.
y = (4x + 3)⁴(x + 1)⁻³
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Ch. 3 - Derivatives
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 3, Problema 3.8.10
If r + s² + v³ = 12, dr/dt = 4, and ds/dt = –3, find dv/dt when r = 3 and s = 1.
Guida verificata passo dopo passo1
Start by differentiating the given equation with respect to time t: \( \frac{d}{dt}(r + s^2 + v^3) = \frac{d}{dt}(12) \).
Apply the chain rule to differentiate each term: \( \frac{dr}{dt} + 2s \frac{ds}{dt} + 3v^2 \frac{dv}{dt} = 0 \).
Substitute the given values into the differentiated equation: \( 4 + 2(1)(-3) + 3v^2 \frac{dv}{dt} = 0 \).
Simplify the equation: \( 4 - 6 + 3v^2 \frac{dv}{dt} = 0 \), which simplifies to \( -2 + 3v^2 \frac{dv}{dt} = 0 \).
Solve for \( \frac{dv}{dt} \) by isolating it: \( 3v^2 \frac{dv}{dt} = 2 \), then \( \frac{dv}{dt} = \frac{2}{3v^2} \). Substitute the value of v when r = 3 and s = 1 to find \( \frac{dv}{dt} \).
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Implicit Differentiation
Implicit differentiation is a technique used to find the derivative of a function when it is not explicitly solved for one variable in terms of another. In this problem, the equation r + s² + v³ = 12 involves multiple variables, and we need to differentiate with respect to time t to find the rate of change of v, denoted as dv/dt.
Video consigliato:
Percorso guidato
05:14
Finding The Implicit Derivative
Chain Rule
The chain rule is a fundamental principle in calculus used to differentiate composite functions. It is essential here because we are dealing with functions of multiple variables that change with respect to time. For instance, when differentiating s² with respect to t, we apply the chain rule: d(s²)/dt = 2s * (ds/dt).
Video consigliato:
05:02Intro to the Chain Rule
Substitution of Known Values
After differentiating the equation, substituting known values is crucial to solve for the unknown rate of change. In this problem, we substitute r = 3, s = 1, dr/dt = 4, and ds/dt = -3 into the differentiated equation to find dv/dt. This step simplifies the equation, allowing us to isolate and solve for the desired rate.
Video consigliato:
04:27Substitution With an Extra Variable
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