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CalculusCalculus: An Applied Approach (MindTap Course List)10th Edition

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Chapter 7.4, Problem 52E
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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Finding Second Partial Derivatives In Exercises 45-52, find the four second partial derivatives. See Example 6.

z = x x + y

To determine

To calculate: The second partial derivatives for the function z=xx+y.

Explanation

Given information:

The provided function is z=xx+y.

Formula used:

Consider the function z=f(x,y) then for the value of ∂z∂x consider y to be constant and differentiate with respect to x and the value of ∂z∂y consider x to be constant and differentiate with respect to y.

According to Higher-Order Partial Derivatives,

∂∂x(∂f∂x)=∂2f∂x2=fxx∂∂y(∂f∂y)=∂2f∂y2=fyy∂∂y(∂f∂x)=∂2f∂y∂x=fxy∂∂x(∂f∂y)=∂2f∂x∂y=fyx

Calculation:

Consider the provided function is,

z=xx+y

Partially derivative of the function z=xx+y with respect to x.

zx=∂∂x(xx+y)=(x+y)∂∂x(x)−(x)∂∂x(x+y)(x+y)2=(x+y)(1)−(x)(1)(x+y)2=y(x+y)2

Partially derivative of the function z=xx+y with respect to y.

zy=∂∂y(xx+y)=(x+y)∂∂y(x)−(x)∂∂y(x+y)(x+y)2=(x+y)(0)−(x)(1)(x+y)2=−x(x+y)2

Again, partially derivative of the function zx=y(x+y)2 with respect to x.

zxx=∂∂x(y(x+y)2)=y∂∂x((x+y)−2)=y(x+y)3⋅(−2)=−2y(x+y)3

Again, partially derivative of the function zx=y(x+y)2 with respect to y

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Sect-7.1 P-1CPSect-7.1 P-2CPSect-7.1 P-3CPSect-7.1 P-4CPSect-7.1 P-5CPSect-7.1 P-6CPSect-7.1 P-7CPSect-7.1 P-1SWUSect-7.1 P-2SWUSect-7.1 P-3SWU
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Sketch the region enclosed by the given curves and calculate its area. y=2xx2,y=0

Calculus (MindTap Course List)

In Exercises 19 and 20, determine whether the lines AB and CD are parallel. 19. A(1, 2), B(3, 10) and C(1, 5), ...

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

Critical Thinking Numbers are often assigned to data that are categorical in nature. (a) Consider these number ...

Understanding Basic Statistics

a. After 6 points have been added to every score in a sample the mean is found to be M = 10 and the standard de...

Statistics for The Behavioral Sciences (MindTap Course List)

A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a funct...

Single Variable Calculus

For Problems 2-10, state the property of equality or the property of real numbers that justifies each of the st...

Intermediate Algebra

Evaluate expressions in Exercises 3756, rounding your answer to four significant digits where necessary. 169

Finite Mathematics

Let A denote the percentage of one constituent in a randomly selected rock specimen, and let B denote the perce...

Probability and Statistics for Engineering and the Sciences

Fibonacci Sums Make a conjecture for each of the following sums, where Fn represent the nth Fibonacci number. a...

Mathematical Excursions (MindTap Course List)

In Problems 45-48, determine the missing factor. 46.

Mathematical Applications for the Management, Life, and Social Sciences

After injection of a dose D of insulin, the concentration of insulin in a patients system decays exponentially ...

Single Variable Calculus: Early Transcendentals

Prove the following identities. cos4=14+cos22+cos224

Trigonometry (MindTap Course List)

True or False Label each of the following statements as either true or false. Whether or not a given polynomial...

Elements Of Modern Algebra

Which is larger? 1 kW or 1 MW

Elementary Technical Mathematics

In Exercises 29 to 31, M and Nare the midpoints of sides RS and RT ofRST, respectively. Given: MN=x2+5 ST=x2x+5...

Elementary Geometry For College Students, 7e

Adam Marsh is purchasing a $134,000 condominium apartment. The down payment is 20. and the balance will be fina...

Contemporary Mathematics for Business & Consumers

Continuity of a Composite Function In Exercises 65-70, discuss the continuity of the composite function h(x)=f(...

Calculus of a Single Variable

Completing the Square In Exercises 33-36, find the Indefinite integral xx2+6x+12dx

Calculus: Early Transcendental Functions (MindTap Course List)

In Exercise 71-80, determine whether the statement is true or false. If it is true explain why it is true. If i...

Finite Mathematics for the Managerial, Life, and Social Sciences

Note: Exercises preceded by an asterisk are of a more challenging nature. In Exercises 3 to 8, use the drawing ...

Elementary Geometry for College Students

HOW DO YOU SEE IT? The graph shows the flow rate of water at a pumping station for one day. (a) Approximate the...

Calculus (MindTap Course List)

Finding a Unit Vector In Exercises 35-38, find the unit vector in the direction of v and verify that it has len...

Multivariable Calculus

Sometimes, Always, or Never: If an ≥ bn ≥ 0 for all n and {bn} diverges, then {an} diverges.

Study Guide for Stewart's Multivariable Calculus, 8th

True or False: The function y = f(x) graphed at the right could be a solution to y = x + y.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

Linear and Quadratic Approximations The linear and quadratic approximations of a function f at x=a are P1(x)=f(...

Calculus

A sample of 16 items provides a sample standard deviation of 9.5. Test the following hypotheses using = .05. W...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Distinguish between a hypothesis and a prediction.

Research Methods for the Behavioral Sciences (MindTap Course List)

Five observations taken for two variables follow. a. Develop a scatter diagram for these data. b. What does the...

Statistics for Business & Economics, Revised (MindTap Course List)

The following data on sale price, size, and land-to-building ratio for 10 large industrial properties appeared ...

Introduction To Statistics And Data Analysis

Give an example of a situation (aside from gender) in which a researcher must examine preexisting groups.

Research Methods for the Behavioral Sciences (MindTap Course List)

Find integer bounds for the zeros of each polynomial function. Answers can vary. P(x)=x5x45x3+5x2+4x5

College Algebra (MindTap Course List)

For each of the values in the following table, the true value and measured value are given. Determine the absol...

Mathematics For Machine Technology

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