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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.8.18b
Chapter 3, Problem 3.8.18b

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?

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1
To find how ds/dt is related to dy/dt and dz/dt, we start by differentiating the given formula for the diagonal length s with respect to time t. The formula is s = √(x² + y² + z²).
Since x is constant, its derivative with respect to time, dx/dt, is zero. Therefore, we only need to consider the derivatives of y and z with respect to time.
Apply the chain rule to differentiate s with respect to t: ds/dt = (1/2) * (1/√(x² + y² + z²)) * (2y * dy/dt + 2z * dz/dt).
Simplify the expression: ds/dt = (y * dy/dt + z * dz/dt) / √(x² + y² + z²).
This equation shows how the rate of change of the diagonal length, ds/dt, is related to the rates of change of the edges y and z, dy/dt and dz/dt, respectively, when x is constant.

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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 are used to find the rate of change of a function with respect to one variable while keeping other variables constant. In this problem, since x is constant, we focus on how s changes with respect to y and z, requiring the use of partial derivatives to express ds/dt in terms of dy/dt and dz/dt.
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Derivatives

Chain Rule

The chain rule in calculus is a formula for computing the derivative of a composite function. It is essential here to relate ds/dt to dy/dt and dz/dt, as it allows us to differentiate s = √(x² + y² + z²) with respect to time t, considering y and z as functions of t.
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Intro to the Chain Rule

Pythagorean Theorem in 3D

The formula s = √(x² + y² + z²) is derived from the Pythagorean theorem extended to three dimensions, representing the diagonal of a rectangular box. Understanding this geometric interpretation helps in visualizing how changes in y and z affect the diagonal length s, which is crucial for setting up the relationship between ds/dt, dy/dt, and dz/dt.
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Fundamental Theorem of Calculus Part 1
Related Practice
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)

Textbook Question

The radius r of a circle is measured with an error of at most 2%. What is the maximum corresponding percentage error in computing the circle’s


b. area?

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

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.


x√(1 + 2y) + y = x², P(1,0)

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).

c. How are dx/dt, dy/dt, and dz/dt related if s is constant?

Textbook Question

Area The area A of a triangle with sides of lengths a and b enclosing an angle of measure θ is

A = (1/2) ab sinθ.


b. How is dA/dt related to dθ/dt and da/dt if only b is constant?