Textbook Question
If x = y³ – y and dy/dt = 5, then what is dx/dt when y = 2?
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 3.7.16
Find dr/dθ in Exercises 15–18.
r – 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴
Verified step by step guidance1
Start by identifying the given equation: r - 2√θ = (3/2)θ²/³ + (4/3)θ³/⁴. We need to find dr/dθ, which is the derivative of r with respect to θ.
Rearrange the equation to express r explicitly in terms of θ: r = 2√θ + (3/2)θ²/³ + (4/3)θ³/⁴.
Differentiate each term of the equation with respect to θ. For the term 2√θ, use the derivative of θ^(1/2), which is (1/2)θ^(-1/2).
For the term (3/2)θ²/³, apply the power rule: the derivative is (3/2) * (2/3)θ^(2/3 - 1). Simplify this expression.
For the term (4/3)θ³/⁴, again use the power rule: the derivative is (4/3) * (3/4)θ^(3/4 - 1). Simplify this expression. Combine all the derivatives to find dr/dθ.
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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, r is given implicitly in terms of θ, so we differentiate both sides of the equation with respect to θ, treating r as a function of θ, and then solve for dr/dθ.
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05:14
Finding The Implicit Derivative
Power Rule for Differentiation
The power rule is a basic rule in calculus used to differentiate functions of the form x^n, where n is a real number. The derivative of x^n is n*x^(n-1). In this problem, terms like θ^(2/3) and θ^(3/4) require applying the power rule to find their derivatives with respect to θ.
Recommended video:
04:02The Power Rule
Chain Rule
The chain rule is used to differentiate composite functions. It states that the derivative of a composite function f(g(x)) is f'(g(x)) * g'(x). In this context, when differentiating terms involving θ raised to a power, the chain rule helps in handling the differentiation of nested functions, especially when θ is under a radical or fractional exponent.
Recommended video:
05:02Intro to the Chain Rule
Related Practice
Textbook Question
In Exercises 41–58, find dy/dt.
y = (1 + tan⁴(t/12))³
1
views
Textbook Question
Derivative Calculations
In Exercises 1–8, given y = f(u) and u = g(x), find dy/dx = f'(g(x)) g'(x).
y = √u, u = sin x
Textbook Question
a. Graph the function
ƒ(x) = { x, -1 ≤ x < 0
{ tan x, 0 ≤ x ≤ π/4.
b. Is ƒ continuous at x = 0?
c. Is ƒ differentiable at x = 0?
Give reasons for your answers.
Textbook Question
Derivatives in Differential Form
In Exercises 17–28, find dy.
y = sec(x² − 1)
Textbook Question
Is there a value of c that will make
f(x) = { (sin²(3x)) / x², x ≠ 0
c, x = 0
continuous at x = 0? Give reasons for your answer.