Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
y = x³/3 + x²/2 + x/4
Ch. 3 - Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 3, Problem 3.8.5
If y = x² and dx/dt = 3, then what is dy/dt when x = –1?
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
Recommended video:
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.
Recommended video:
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.
Recommended video:
04:16Intro To Related Rates
Related Practice
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
g(t) = (1 + sin(3t) / (3 − 2t))⁻¹
Textbook Question
Find the derivatives of the functions in Exercises 19–40.
y = (5 − 2x)⁻³ + (1 / 8)(2 / x + 1)⁴
Textbook Question
Derivative Calculations
In Exercises 1–12, find the first and second derivatives.
w = 3z⁷ − 7z³ + 21z²
Textbook Question
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
Textbook Question
Finding Linearizations
In Exercises 1–5, find the linearization L(x) of f(x) at x = a.
f(x) = ∛x, a = −8