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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 2 y 2 2 x y

To determine

To calculate: The second partial derivatives for the function z=x2y22xy.

Explanation

Given information:

The provided function is z=x2y22xy.

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=x2y22xy

Partially derivative of the function z=x2y22xy with respect to x.

zx=x(x2y22xy)=2xyx(x2y2)(x2y2)x(2xy)(2xy)2=2xy(2x)(x2y2)(2y)(2xy)2=x2+y22x2y

Partially derivative of the function z=x2y22xy with respect to y.

zy=y(x2y22xy)=2xyy(x2y2)(x2y2)y(2xy)(2xy)2=2xy(2x)(x2y2)(2x)(2xy)2=x2+y22xy2

Again, partially derivative of the function zx=x2+y22x2y with respect to x.

zxx=x(x2+y22x2y)=2x2yx(x2+y2)(x2+y2)x(2x2y)(2x2y)2=2x2y(2x)(x2+y2)(4xy)(2x2y)2=yx3

Again, partially derivative of the function zx=x2+y22x2y with respect to y

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