   Chapter 7.4, Problem 50E ### 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 = 9 − x 2 − y 2

To determine

To calculate: The second partial derivatives for the function z=9x2y2.

Explanation

Given information:

The provided function is z=9x2y2.

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=9x2y2

Partially derivative of the function z=9x2y2 with respect to x.

zx=x(9x2y2)=12(9x2y2)12x(9x2y2)=129x2y2(2x)=x9x2y2

Partially derivative of the function z=9x2y2 with respect to y.

zy=y(9x2y2)=12(9x2y2)12y(9x2y2)=129x2y2(2y)=y9x2y2

Again, partially derivative of the function zx=x9x2y2 with respect to x.

zxx=x(x9x2y2)=9x2y2x(x)xx(9x2y2)(9x2y2)2=9x2y2(1)x(x9x2y2)9x2y2=y29(9x2y2)32

Again, partially derivative of the function zx=x9x2y2 with respect to y

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