Explanation Given: The function is z = x ln | x y | . Definition used: Partial derivatives: “The partial derivative of f with respect to x is the derivative of f obtained by treating x as a variable and y as a constant. The partial derivative of f with respect to y is the derivative of f obtained by treating y as a variable and x as a constant.” Calculation: The function z = x ln | x y | . Treat y as a constant and x as a variable to find the Partial derivative of the function z ( x , y ) with respect to x . z x = ∂ ∂ x ( x ln | x y | ) = ( x ) ∂ ∂ x ( ln | x y | ) + ( ln | x y | ) ∂ ∂ x ( x ) = ( x ) ( ∂ ∂ ( | x y | ) ( ln | x y | ) ∂ ∂ x ( x y ) ) + ( ln | x y | ) ∂ ∂ x ( x ) = ( x ) ( 1 x y ( x ∂ ∂ x ( y ) + ( y ) ∂ ∂ x ( x ) ) ) + ( ln | x y | ) ∂ ∂ x ( x ) Simplify the terms and obtain the partial derivatives. z x = ( x ) ( 1 x y ( x ( 0 ) + ( y ) ( 1 ) ) ) + ( ln | x y | ) ( 1 ) = x y x y + ln | x y | = 1 + ln | x y | Thus, the partial derivative with respect to x is, z x = 1 + ln | x y | . Treat y as a constant and x as a variable to find the Partial derivative of the function z x ( x , y ) with respect to x . z x x = ∂ ∂ x ( z x ) = ∂ ∂ x ( 1 + ln | x y | ) = ∂ ∂ x ( 1 ) + ∂ ∂ x ( ln | x y | ) = 0 + ∂ ∂ | | x y | | ( ln | x y | ) ∂ ∂ x ( x y ) Simplify the terms and obtain the partial derivatives. z x x = 1 x y ( ( x ) ∂ ∂ x ( y ) + ( y ) ∂ ∂ x ( x ) ) = 1 x y ( ( x ) ( 0 ) + ( y ) ( 1 ) ) = y x y = 1 x Therefore, the partial derivatives z x x = 1 x . Treat x as a constant and y as a variable to find the Partial derivative of the function z ( x , y ) with respect to y . z y = ∂ ∂ y ( x ln | x y | ) = [ ( x ) ∂ ∂ y ( ln | x y | ) + ( ln | x y | ) ∂ ∂ y ( x ) ] = [ ( x ) ( ∂ ∂ ( x y ) ( ln | x y | ) ∂ ∂ y ( x y ) ) + ( ln | x y | ) ∂ ∂ y ( x ) ] = [ ( x ) ( 1 x y ( ( x ) ∂ ∂ y ( y ) + ( y ) ∂ ∂ y ( x ) ) ) + ( ln | x y | ) ∂ ∂ y ( x ) ] Simplify further and obtain the partial derivatives

Finite Mathematics and Calculus with Applications

1st Edition

ISBN: 9781323188361

Author: Margaret Lial

Publisher: Pearson Education

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Question

Chapter 17.2, Problem 31E

To determine

To find: All second order partial derivatives for given function.

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Chapter 17.2, Problem 30E

Chapter 17.2, Problem 32E

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Chapter 17 Solutions

Finite Mathematics and Calculus with Applications

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If the function...Ch. 17 - Prob. 85RE Ch. 17 - Prob. 86RE Ch. 17 - Prob. 87RE Ch. 17 - Prob. 88RE Ch. 17 - Prob. 89RE Ch. 17 - Prob. 90RE Ch. 17 - Prob. 91RE Ch. 17 - Prob. 92RE Ch. 17 - Prob. 93RE Ch. 17 - Prob. 94RE Ch. 17 - Prob. 95RE Ch. 17 - Prob. 96RE Ch. 17 - Profit The total profit from 1 acre of a certain...Ch. 17 - Prob. 98RE Ch. 17 - Prob. 99RE Ch. 17 - Prob. 100RE Ch. 17 - Prob. 101RE Ch. 17 - Prob. 102RE Ch. 17 - Prob. 103RE Ch. 17 - Prob. 104RE Ch. 17 - Prob. 105RE Ch. 17 - Prob. 106RE

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All second order partial derivatives for given function.

Solution Summary: The author explains the second order partial derivatives of the function z=xmathrmlnleft|xy

To find: All second order partial derivatives for given function.

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Chapter 17 Solutions