   Chapter 14, Problem 20RE

Chapter
Section
Textbook Problem

Find all second partial derivatives of f.20. z = xe−2y

To determine

To find: The second order partial derivative of the function z(x,y)=xe2y .

Explanation

Given:

The function is, z(x,y)=xe2y .

Calculation:

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

zx=x(xe2y)=e2yx(x)=e2y(1)

zx=e2y (1)

Differentiate the equation (1) with respect to x and obtain the second order derivative, zxx

2zx2=x(e2y)=0

Thus, zxx(x,y)=0 .

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

zy=y(xe2y)=xy(e2y)=xe2y(2)

zy=2xe2y (2)

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

2zy2=y(2xe2y)=2xy(e2y)=2xe2y(2)=4xe2y

Hence, zyy(x,y)=4xe2y

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