   Chapter 7.4, Problem 6CP ### 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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# Find the second partial derivatives of f ( x , y ) = 4 x 2 y 2 + 2 x + 4 y 2 .

To determine

To calculate: The second partial derivatives for the function f(x,y)=4x2y2+2x+4y2.

Explanation

Given information:

The provided function is f(x,y)=4x2y2+2x+4y2.

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,

f(x,y)=4x2y2+2x+4y2

Partially derivative of the function f(x,y)=4x2y2+2x+4y2 with respect to x.

fx(x,y)=x(4x2y2+2x+4y2)=4y2x(x2)+2x(x)+4y2x(1)=8xy2+2

Partially derivative of the function f(x,y)=4x2y2+2x+4y2 with respect to y.

fy(x,y)=y(4x2y2+2x+4y2)=4x2y(y2)+2xy(1)+4y(y2)=8x2y+8y

Again, partially derivative of the function fx(x,y)=8xy2+2 with respect to x

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