   Chapter 14, Problem 19RE

Chapter
Section
Textbook Problem

Find all second partial derivatives of f.19. f(x, y) = 4x3 − xy2

To determine

To find: The second order partial derivative of the function f(x,y)=4x3xy2 .

Explanation

Given:

The function is, f(x,y)=4x3xy2 .

Calculation:

Differentiate the given function with respect to x and obtain fx .

fx=x(4x3xy2)=4(3x2)y2(1)

fx=12x2y2 (1)

Differentiate the equation (1) with respect to x and obtain the second order derivative of the function, fxx .

2fx2=x(12x2y2)=12(2x)0=24x

Thus, fxx(x,y)=24x .

Differentiate the given function with respect to y and obtain fy .

fy=y(4x3xy2)=0x(2y)

fy=2xy (2)

Differentiate the equation (2) with respect to y and obtain the second order derivative of the function, fyy .

2fy2=y(2xy)=2x(1)=2x

Hence, fyy(x,y)=2x

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