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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 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=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)=yx((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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