Domanda del libro di testo
Find the derivatives of the functions in Exercises 19–40.
g(t) = (1 + sin(3t) / (3 − 2t))⁻¹
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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 3.8.5
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
Guida verificata passo dopo passo1
First, identify the given function: y = x². This is the function that relates y and x.
Recognize that you need to find dy/dt, which is the rate of change of y with respect to time t.
Use the chain rule for differentiation, which states that dy/dt = (dy/dx) * (dx/dt).
Differentiate y = x² with respect to x to find dy/dx. The derivative of x² with respect to x is 2x.
Substitute the given values into the chain rule formula: dy/dt = 2x * (dx/dt). Use x = -1 and dx/dt = 3 to find dy/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, it helps us differentiate y = x² with respect to time t, considering x as a function of t, to find dy/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 states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. Here, it allows us to relate dy/dt to dx/dt by differentiating y = x² with respect to t.
Video consigliato:
05:02Intro to the Chain Rule
Related Rates
Related rates problems involve finding the rate at which one quantity changes with respect to time, given the rate of change of another related quantity. In this scenario, we are given dx/dt and need to find dy/dt when x = -1, using the relationship between x and y provided by the equation y = x².
Video consigliato:
Percorso guidato
04:16Intro To Related Rates
Pratica correlata
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Find the derivatives of the functions in Exercises 19–40.
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Domanda del libro di testo
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In Exercises 1–12, find the first and second derivatives.
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In Exercises 29 and 30, find the slope of the curve at the given points.
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Approximation Error
In Exercises 29–34, each function f(x) changes value when x changes from x₀ to x₀ + dx. Find
a. the change Δf = f(x₀ + dx) − f(x₀);
b. the value of the estimate df = fʹ(x₀) dx; and
c. the approximation error |Δf − df|.
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f(x) = x² + 2x, x₀ = 1, dx = 0.1
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Domanda del libro di testo
One-Sided Derivatives
Compute the right-hand and left-hand derivatives as limits to show that the functions in Exercises 37–40 are not differentiable at the point P.
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