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

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Chapter 7, Problem 55RE
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Ron Larson
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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 55-60, find the second partial derivatives.

f ( x , y ) = 3 x 2 − x y + 2 y 3

To determine

To calculate: The second partial derivative of function f(x,y)=3x2−xy+2y3.

Explanation

Given Information:

The provided function is f(x,y)=3x2−xy+2y3.

Formula used:

The power rule of differentiation with respect to x by holding y constant,

∂∂x[f(x,y)]n=n[f(x,y)]n−1⋅∂∂xf(x,y)

The power rule of differentiation with respect to y by holding x constant,

∂∂y[f(x,y)]n=n[f(x,y)]n−1⋅∂∂yf(x,y)

Calculation:

Consider the function,

f(x,y)=3x2−xy+2y3

Differentiate with respect to x by holding y constant,

∂∂xf(x,y)=∂∂x(3x2−xy+2y3)=∂∂x(3x2)+∂∂x(−xy)+∂∂x(2y3)=6x−y

The first partial derivative of function f(x,y)=3x2−xy+2y3 with respect to x,

∂∂xf(x,y)=6x−y

Now again differentiate with respect to x by holding y constant,

∂2∂2xf(x,y)=∂∂x(6x−y)=6

Differentiate with respect to y by holding x constant,

∂∂yf(x,y)=∂∂y(3x2−xy+2y3)=∂∂y(3x2)+∂∂y(−xy)+∂∂y(2y3)=−x+6y2

The first partial derivative of function f

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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
Sect-7.1 P-4SWUSect-7.1 P-5SWUSect-7.1 P-6SWUSect-7.1 P-7SWUSect-7.1 P-8SWUSect-7.1 P-1ESect-7.1 P-2ESect-7.1 P-3ESect-7.1 P-4ESect-7.1 P-5ESect-7.1 P-6ESect-7.1 P-7ESect-7.1 P-8ESect-7.1 P-9ESect-7.1 P-10ESect-7.1 P-11ESect-7.1 P-12ESect-7.1 P-13ESect-7.1 P-14ESect-7.1 P-15ESect-7.1 P-16ESect-7.1 P-17ESect-7.1 P-18ESect-7.1 P-19ESect-7.1 P-20ESect-7.1 P-21ESect-7.1 P-22ESect-7.1 P-23ESect-7.1 P-24ESect-7.1 P-25ESect-7.1 P-26ESect-7.1 P-27ESect-7.1 P-28ESect-7.1 P-29ESect-7.1 P-30ESect-7.1 P-31ESect-7.1 P-32ESect-7.1 P-33ESect-7.1 P-34ESect-7.1 P-35ESect-7.1 P-36ESect-7.1 P-37ESect-7.1 P-38ESect-7.1 P-39ESect-7.1 P-40ESect-7.1 P-41ESect-7.1 P-42ESect-7.1 P-43ESect-7.1 P-44ESect-7.2 P-1CPSect-7.2 P-2CPSect-7.2 P-3CPSect-7.2 P-1SWUSect-7.2 P-2SWUSect-7.2 P-3SWUSect-7.2 P-4SWUSect-7.2 P-5SWUSect-7.2 P-6SWUSect-7.2 P-1ESect-7.2 P-2ESect-7.2 P-3ESect-7.2 P-4ESect-7.2 P-5ESect-7.2 P-6ESect-7.2 P-7ESect-7.2 P-8ESect-7.2 P-9ESect-7.2 P-10ESect-7.2 P-11ESect-7.2 P-12ESect-7.2 P-13ESect-7.2 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Convert the expressions in Exercises 6584 to power form. xx

Finite Mathematics and Applied Calculus (MindTap Course List)

A researcher is curious about the average distance traveled by Canada geese during peak fall migration in the s...

Essentials of Statistics for The Behavioral Sciences (MindTap Course List)

You are the governor of the state and must decide which of four metropolitan police departments will win an awa...

Essentials Of Statistics

112 Find the area of the surface. The part of the plane 6x+4y+2z=1 that lies inside the cylinder x2+y2=25

Calculus (MindTap Course List)

Market Equilibrium The monthly demand and supply functions for the Luminar desk lamp are given by p = d(x) = 1....

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

cores on a standardized reading test for 4th-grade students form a normal distribution with =60 and =20 . What ...

Statistics for The Behavioral Sciences (MindTap Course List)

Terminology Which graphical display shows each data value (or truncated data value) in order from smallest to l...

Understanding Basic Statistics

Sketch the graph of the function. 46. f(x) = |x + 2|

Single Variable Calculus

Verify that for A and B as defined in Exercises 1 to 10, n(A U B) = n(A) + n(B) - n(A B).

Mathematical Excursions (MindTap Course List)

For Problems 1-10, identify each statement as true or false. Objective 1 All whole numbers are integers.

Intermediate Algebra

Find the value of k such that (1,k) is equidistant from (0,0) and (2,1).

Finite Mathematics

The article Effects of Short-Term Warming on Low and High latitude Forest Ant Communities (Ecoshpere, May 2011,...

Probability and Statistics for Engineering and the Sciences

In problems 37-44, graph the subset of the real numbers that is represented by each of the following and write...

Mathematical Applications for the Management, Life, and Social Sciences

Find the Taylor series of f(x) = sin x at a = /6.

Single Variable Calculus: Early Transcendentals

According to the Financial Times, in a recent recession, outstanding consumer credit in the United States fell ...

Contemporary Mathematics for Business & Consumers

Use Figure 5.19 to prove Theorem 5.4.2. Theorem 5.4.2 The length of the altitude to the hypotenuse of a right t...

Elementary Geometry For College Students, 7e

The resistance, R, in an electrical circuit varies according to the equation R=1.5V, where V is in volts. Graph...

Elementary Technical Mathematics

Sketch the graph of y=12sin(x). Which of the following matches your graph?

Trigonometry (MindTap Course List)

Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.

Elements Of Modern Algebra

Graphs of Inverse Functions Have Reciprocal Slopes In Exercises 13-16, show that the slopes or the graphs of f ...

Calculus: Early Transcendental Functions (MindTap Course List)

Finding a Derivative In Exercises 9-20, use the rules of differentiation to find the derivative of the function...

Calculus of a Single Variable

Note: Exercises preceded by an asterisk are of a more challenging nature. Given: OVRSinOOV=9andOT=6 Find: RS

Elementary Geometry for College Students

LIFE EXPECTANCY Figures for life expectancy at birth of U.S. residents in 2012 were 81.2, 83.9, and, 78.1 years...

Finite Mathematics for the Managerial, Life, and Social Sciences

Verifying that a Series Converges In Exercises 21-24, use the power series solution of the differential equatio...

Multivariable Calculus

Propulsion Neglecting air resistance and the weight of the propellant, determine the work done in propelling a ...

Calculus (MindTap Course List)

What substitution should be made to evaluate x2+2x(x+1)dx? a) u=x2+2x b) u = x2 + 2x c) u = x + 1 d) u=x2+2x(x+...

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

For

Study Guide for Stewart's Multivariable Calculus, 8th

Finding an Indefinite Integral In Exercises 21-26, find the indefinite integral. sinx1+cosxdx

Calculus

The mean preparation fee HR Block charged retail customers last year was 183 (The Wall Street Journal, March 7,...

STATISTICS F/BUSINESS+ECONOMICS-TEXT

Reduce to halves. a. 2510 b. 1812 c. 12636 d. 22550

Mathematics For Machine Technology

A Harris Poll press release dated November 1, 2016, summarized results of a survey of 2463 adults and 510 teens...

Introduction To Statistics And Data Analysis

In addition to the key words, you should also be able to define each of the following terms: Publication Manual...

Research Methods for the Behavioral Sciences (MindTap Course List)

Can a research study be an experiment without a control group? Can a study be an experiment without controlling...

Research Methods for the Behavioral Sciences (MindTap Course List)

Getting Ready You should be able to complete these vocabulary and concept statements before you proceed to the ...

College Algebra (MindTap Course List)

The Scholastic Aptitude Test (SAT) consists of three parts: critical reading, mathematics, and writing. Each pa...

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

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