Explanation Given : The function is f ( x , y ) = ln ( 1 + x 2 y 2 ) . Calculation : The partial derivative of f ( x , y ) with respect to x is, f x ( x , y ) = ∂ ∂ x [ ln ( 1 + x 2 y 2 ) ] = 1 1 + x 2 y 2 ∂ ∂ x ( 1 + x 2 y 2 ) = 1 1 + x 2 y 2 ∂ ∂ x ( x 2 y 2 ) = y 2 1 + x 2 y 2 ∂ ∂ x ( x 2 ) = 2 x y 2 1 + x 2 y 2 The partial derivative of f x ( x , y ) with respect to x is, f x ( x , y ) = 2 x y 2 1 + x 2 y 2 f x x ( x , y ) = ∂ ∂ x ( 2 x y 2 1 + x 2 y 2 ) = 2 ( 1 + x 2 y 2 ) ∂ ∂ x ( x y 2 ) − x y 2 ∂ ∂ x ( 1 + x 2 y 2 ) ( 1 + x 2 y 2 ) 2 f x x ( x , y ) = 2 ( 1 + x 2 y 2 ) 2 [ ( 1 + x 2 y 2 ) y 2 − x y 2 ( y 2 2 x ) ] Simplify further as follows. f x x ( x , y ) = 2 ( 1 + x 2 y 2 ) 2 [ ( 1 + x 2 y 2 ) y 2 − x y 2 ( y 2 2 x ) ] = 2 ( 1 + x 2 y 2 ) 2 [ y 2 + x 2 y 4 − 2 x 2 y 4 ] f x x ( x , y ) = 2 y 2 ( 1 + x 2 y 2 ) 2 ( 1 − x 2 y 2 ) The partial derivative of f ( x , y ) with respect to y is, f ( x , y ) = ln ( 1 + x 2 y 2 ) f y ( x , y ) = ∂ ∂ y [ ln ( 1 + x 2 y 2 ) ] = 1 1 + x 2 y 2 ∂ ∂ y ( 1 + x 2 y 2 ) f y ( x , y ) = 1 1 + x 2 y 2 ∂ ∂ y ( x 2 y 2 ) Simplify further as follows. f y ( x , y ) = 1 1 + x 2 y 2 ∂ ∂ y ( x 2 y 2 ) = x 2 1 + x 2 y 2 ∂ ∂ x ( y 2 ) f y ( x , y ) = 2 x 2 y 1 + x 2 y 2 The partial derivative of f y ( x , y ) with respect to y is, f y ( x , y ) = 2 x 2 y 1 + x 2 y 2 f y y ( x , y ) = ∂ ∂ y ( 2 x 2 y 1 + x 2 y 2 ) = 2 ( 1 + x 2 y 2 ) ∂ ∂ y ( x 2 y ) − x 2 y ∂ ∂ y ( 1 + x 2 y 2 ) ( 1 + x 2 y 2 ) 2 f y y ( x , y ) = 2 ( 1 + x 2 y 2 ) 2 [ ( 1 + x 2 y 2 ) x 2 − x 2 y ( x 2 2 y ) ] Simplify further as follows

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

10th Edition

ISBN: 9781305657861

Author: Soo T. Tan

Publisher: Cengage Learning

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Chapter 8.2, Problem 42E

To determine

To find: The second-order partial derivatives of the function and to show that the mixed partial derivatives fxy and fyx are equal.

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Chapter 8.2, Problem 41E

Chapter 8.2, Problem 43E

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences (MindTap Course List)

Ch. 8.1 - What is a function of two variables? Give an...Ch. 8.1 - Prob. 2CQ Ch. 8.1 - Prob. 3CQ Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Let f(x, y) = x2+ 2xy x + 3. Compute f(1, 2),...Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Let h(s, t) = s ln t t ln s. Compute h(1, e),...

Ch. 8.1 - Prob. 8E Ch. 8.1 - Let g(r, s, t) = res/t. Compute g(1, 1, 1), g(1,...Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 14E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 16E Ch. 8.1 - In Exercises 11-18, find the domain of the...Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - In Exercises 19-24, sketch the level curves of the...Ch. 8.1 - Prob. 22E Ch. 8.1 - In Exercises 1924, sketch the level curves of the...Ch. 8.1 - Prob. 24E Ch. 8.1 - Find an equation of the level curve of f (x, y) =...Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Prob. 31E Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - POISEITLLES LAW Poiseuilles Law states that the...Ch. 8.1 - COST FUNCTION FOR A LOUDSPEAKER SYSTEM Acrosonic...Ch. 8.1 - Prob. 38E Ch. 8.1 - REVENUE FUNCTIONS FOR DESKS Country Workshop...Ch. 8.1 - Prob. 40E Ch. 8.1 - 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Prob. 1TE Ch. 8.4 - Prob. 2TE Ch. 8.4 - Prob. 3TE Ch. 8.4 - Prob. 4TE Ch. 8.4 - Prob. 5TE Ch. 8.4 - Prob. 6TE Ch. 8.4 - Prob. 7TE Ch. 8.4 - Prob. 8TE Ch. 8.4 - Prob. 9TE Ch. 8.5 - What is a constrained relative extremum of a...Ch. 8.5 - Prob. 2CQ Ch. 8.5 - Prob. 1E Ch. 8.5 - Prob. 2E Ch. 8.5 - Prob. 3E Ch. 8.5 - Prob. 4E Ch. 8.5 - Prob. 5E Ch. 8.5 - Prob. 6E Ch. 8.5 - Prob. 7E Ch. 8.5 - Prob. 8E Ch. 8.5 - Prob. 9E Ch. 8.5 - Prob. 10E Ch. 8.5 - In Exercises 1-16. use the method of Lagrange...Ch. 8.5 - Prob. 12E Ch. 8.5 - In Exercises 1-16. use the method of Lagrange...Ch. 8.5 - Prob. 14E Ch. 8.5 - Prob. 15E Ch. 8.5 - Prob. 16E Ch. 8.5 - MAXIMIZING PROFIT The total weekly profit (in...Ch. 8.5 - Prob. 18E Ch. 8.5 - Prob. 19E Ch. 8.5 - Prob. 20E Ch. 8.5 - PACKAGING Find the dimensions of an open...Ch. 8.5 - Prob. 22E Ch. 8.5 - Prob. 23E Ch. 8.5 - Prob. 24E Ch. 8.5 - MINIMIZING CONTAINER COSTS The Betty Moore Company...Ch. 8.5 - Prob. 26E Ch. 8.5 - Prob. 27E Ch. 8.5 - Prob. 28E Ch. 8.5 - Prob. 29E Ch. 8.5 - Prob. 30E Ch. 8.5 - Prob. 31E Ch. 8.5 - Prob. 33E Ch. 8.5 - Prob. 34E Ch. 8.5 - Prob. 36E Ch. 8.5 - Prob. 38E Ch. 8.5 - Prob. 39E Ch. 8.5 - Prob. 40E Ch. 8.5 - Prob. 41E Ch. 8.5 - Prob. 42E Ch. 8.6 - Prob. 1CQ Ch. 8.6 - Prob. 2CQ Ch. 8.6 - Prob. 1E Ch. 8.6 - Prob. 2E Ch. 8.6 - Prob. 3E Ch. 8.6 - Prob. 4E Ch. 8.6 - Prob. 5E Ch. 8.6 - Prob. 6E Ch. 8.6 - Prob. 7E Ch. 8.6 - Prob. 8E Ch. 8.6 - Prob. 9E Ch. 8.6 - Prob. 10E Ch. 8.6 - Prob. 11E Ch. 8.6 - Prob. 12E Ch. 8.6 - Prob. 13E Ch. 8.6 - Prob. 14E Ch. 8.6 - In Exercises 318, find the total differential of...Ch. 8.6 - Prob. 16E Ch. 8.6 - Prob. 17E Ch. 8.6 - Prob. 18E Ch. 8.6 - Prob. 19E Ch. 8.6 - Prob. 20E Ch. 8.6 - Prob. 21E Ch. 8.6 - Prob. 22E Ch. 8.6 - Prob. 23E Ch. 8.6 - Prob. 24E Ch. 8.6 - Prob. 25E Ch. 8.6 - Prob. 26E Ch. 8.6 - Prob. 27E Ch. 8.6 - Prob. 28E Ch. 8.6 - Prob. 29E Ch. 8.6 - Prob. 30E Ch. 8.6 - Prob. 31E Ch. 8.6 - Prob. 32E Ch. 8.6 - Prob. 33E Ch. 8.6 - Prob. 34E Ch. 8.6 - Prob. 35E Ch. 8.6 - Prob. 36E Ch. 8.6 - Prob. 37E Ch. 8.6 - 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Prob. 5E Ch. 8.8 - Prob. 6E Ch. 8.8 - Prob. 7E Ch. 8.8 - Prob. 8E Ch. 8.8 - Prob. 9E Ch. 8.8 - Prob. 10E Ch. 8.8 - Prob. 11E Ch. 8.8 - Prob. 12E Ch. 8.8 - Prob. 13E Ch. 8.8 - Prob. 14E Ch. 8.8 - Prob. 15E Ch. 8.8 - Prob. 16E Ch. 8.8 - Prob. 17E Ch. 8.8 - Prob. 18E Ch. 8.8 - Prob. 19E Ch. 8.8 - Prob. 20E Ch. 8.8 - Prob. 21E Ch. 8.8 - Prob. 22E Ch. 8.8 - Prob. 23E Ch. 8.8 - Prob. 24E Ch. 8.8 - Prob. 25E Ch. 8.8 - Prob. 26E Ch. 8.8 - Prob. 27E Ch. 8.8 - Prob. 28E Ch. 8.8 - Prob. 29E Ch. 8 - Prob. 1CRQ Ch. 8 - Prob. 2CRQ Ch. 8 - Prob. 3CRQ Ch. 8 - Prob. 4CRQ Ch. 8 - Prob. 5CRQ Ch. 8 - Prob. 6CRQ Ch. 8 - Prob. 7CRQ Ch. 8 - Prob. 8CRQ Ch. 8 - Prob. 9CRQ Ch. 8 - Prob. 10CRQ Ch. 8 - Prob. 11CRQ Ch. 8 - Prob. 12CRQ Ch. 8 - Prob. 13CRQ Ch. 8 - Prob. 1RE Ch. 8 - Prob. 2RE Ch. 8 - Prob. 3RE Ch. 8 - Prob. 4RE Ch. 8 - Prob. 5RE Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - Prob. 12RE Ch. 8 - Prob. 13RE Ch. 8 - Prob. 14RE Ch. 8 - 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The second-order partial derivatives of the function and to show that the mixed partial derivatives f x y and f y x are equal.

To find: The second-order partial derivatives of the function and to show that the mixed partial derivatives fxy and fyx are equal.

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Chapter 8 Solutions