   Chapter 7.4, Problem 49E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### 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 = ( 3 x 4 − 2 y 3 ) 3

To determine

To calculate: The second partial derivatives for the function z=(3x42y3)3.

Explanation

Given information:

The provided function is z=(3x42y3)3.

Formula used:

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

According to Higher-Order Partial Derivatives,

x(fx)=2fx2=fxxy(fy)=2fy2=fyyy(fx)=2fyx=fxyx(fy)=2fxy=fyx

Calculation:

Consider the provided function is,

z=(3x42y3)3

Partially derivative of the function z=(3x42y3)3 with respect to x.

zx=x((3x42y3)3)=3(3x42y3)2x(3x42y3)=3(3x42y3)2(12x3)=36x3(3x42y3)2

Partially derivative of the function z=(3x42y3)3 with respect to y.

zy=y((3x42y3)3)=3(3x42y3)2y(3x42y3)=3(3x42y3)2(6y2)=18y2(3x42y3)2

Again, partially derivative of the function zx=36x3(3x42y3)2 with respect to x.

zxx=x(36x3(3x42y3)2)=36x3x((3x42y3)2)+(3x42y3)2x(36x3)=108x2(3x42y3)+(3x42y3)2(108x2)=108x2(3x42y3)(11x42y3)

Again, partially derivative of the function zx=36x3(3x42y3)2 with respect to y

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