Textbook Question
Computer Explorations
Use a CAS to perform the following steps in Exercises 55–62.
b. Using implicit differentiation, find a formula for the derivative dy/dx and evaluate it at the given point P.
2y² + (xy)¹/³ = x² + 2, P(1,1)
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 3.8.18c
Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).
c. How are dx/dt, dy/dt, and dz/dt related if s is constant?
Verified step by step guidance1
Start by understanding the given formula for the diagonal length of a rectangular box: \( s = \sqrt{x^2 + y^2 + z^2} \). This represents the length of the diagonal in terms of the edge lengths x, y, and z.
Since s is constant, differentiate both sides of the equation with respect to time t. This involves using implicit differentiation because x, y, and z are functions of time.
The differentiation of the left side, since s is constant, is \( \frac{ds}{dt} = 0 \).
For the right side, apply the chain rule: \( \frac{d}{dt}(x^2 + y^2 + z^2) = 2x \frac{dx}{dt} + 2y \frac{dy}{dt} + 2z \frac{dz}{dt} \).
Set the derivative of the right side equal to zero (from step 3): \( 2x \frac{dx}{dt} + 2y \frac{dy}{dt} + 2z \frac{dz}{dt} = 0 \). Simplify this equation to find the relationship between \( \frac{dx}{dt}, \frac{dy}{dt}, \) and \( \frac{dz}{dt} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Partial Derivatives
Partial derivatives involve differentiating a function with respect to one variable while keeping other variables constant. In the context of the problem, understanding how each edge length x, y, and z changes over time (dx/dt, dy/dt, dz/dt) requires applying partial derivatives to the formula for the diagonal length s.
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Derivatives
Chain Rule
The chain rule is a fundamental concept in calculus used to differentiate composite functions. It is essential for relating the rates of change of x, y, and z to the rate of change of s. Since s is constant, the chain rule helps establish a relationship between dx/dt, dy/dt, and dz/dt by differentiating the equation s = √(x² + y² + z²) with respect to time.
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05:02Intro to the Chain Rule
Implicit Differentiation
Implicit differentiation is used when a function is not explicitly solved for one variable. In this problem, since s is constant, implicit differentiation allows us to differentiate the equation s = √(x² + y² + z²) with respect to time, treating s as a constant and finding how dx/dt, dy/dt, and dz/dt are interrelated.
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Related Practice
Textbook Question
Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.
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Find the derivatives with respect to x of the following combinations at the given value of x.
c. f(x) / (g(x) + 1), x = 1
Textbook Question
Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.
c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?
Textbook Question
Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.
x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4
Find the first derivatives of the following combinations at the given value of x.
c. ƒ(x) , x = 1
g(x) + 1
Textbook Question
Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).
b. How is ds/dt related to dy/dt and dz/dt if x is constant?
Textbook Question
Particle motion At time t, the position of a body moving along the s-axis is s = t³ − 6t² + 9t m.
c. Find the total distance traveled by the body from t = 0 to t = 2.