Ch. 3 - Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 3 - Derivatives

Problem 64

Chapter 3, Problem 64

If x¹/³ + y¹/³ = 4, find d²y/dx² at the point (8, 8).

Verified step by step guidance

First, differentiate the given equation implicitly with respect to x. The equation is x^(1/3) + y^(1/3) = 4. Differentiating both sides with respect to x gives: d/dx[x^(1/3)] + d/dx[y^(1/3)] = d/dx[4].

Apply the chain rule to differentiate each term. For x^(1/3), the derivative is (1/3)x^(-2/3). For y^(1/3), use the chain rule: (1/3)y^(-2/3) * dy/dx. The derivative of 4 is 0 since it's a constant.

Set up the equation from the derivatives: (1/3)x^(-2/3) + (1/3)y^(-2/3) * dy/dx = 0. Solve for dy/dx to find the first derivative.

To find d²y/dx², differentiate dy/dx with respect to x again. Use implicit differentiation and the chain rule, considering dy/dx as a function of x and y.

Substitute the point (8, 8) into the expressions for dy/dx and d²y/dx² to evaluate the second derivative at this specific point.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 differentiate equations where y is not explicitly solved for x. In this case, we differentiate both sides of the equation x^(1/3) + y^(1/3) = 4 with respect to x, applying the chain rule to the y term. This allows us to find dy/dx, which is essential for further calculations.

Second Derivative

The second derivative, denoted as d²y/dx², measures the rate of change of the first derivative dy/dx. It provides information about the concavity of the function and can indicate points of inflection. To find d²y/dx², we differentiate dy/dx again with respect to x, which may involve using implicit differentiation again.

Chain Rule

The chain rule is a fundamental differentiation rule used when differentiating composite functions. It states that if a function y is dependent on u, which in turn is dependent on x, then the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x. This rule is crucial when differentiating terms involving y in implicit differentiation.

Recommended video:

05:02

Intro to the Chain Rule

Related Practice

Textbook Question

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.

a. How is dS/dt related to dr/dt if h is constant?

238

views

Textbook Question

Temperature and the period of a pendulum For oscillations of small amplitude (short swings), we may safely model the relationship between the period T and the length L of a simple pendulum with the equation:

T = 2π√(L/g),

where g is the constant acceleration of gravity at the pendulum’s location. If we measure g in centimeters per second squared, we measure L in centimeters and T in seconds. If the pendulum is made of metal, its length will vary with temperature, either increasing or decreasing at a rate that is roughly proportional to L. In symbols, with u being temperature and k the proportionality constant,

dL/du = kL.

Assuming this to be the case, show that the rate at which the period changes with respect to temperature is kT/2.

268

views

Textbook Question

[Technology Exercise]

Graph y = 1/(2√x) in a window that has 0 ≤ x ≤ 2. Then, on the same screen, graph

y = (√(x + h) − √x)/h

for h = 1, 0.5, 0.1. Then try h = −1, −0.5, −0.1. Explain what is going on.

180

views

Textbook Question