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 MGRL, LIFE, SOC SCI

10th Edition

ISBN: 9780357465400

Author: Tan

Publisher: CENGAGE L

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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 MGRL, LIFE, SOC SCI

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Ch. 8.1 - Prob. 8E Ch. 8.1 - Let g(r, s, t) = res/t. 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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