Explanation Formula used: Suppose f ( x , y ) is a function of two variables x and y . Then the second order partial derivative of f x with respect to x at the point ( x , y ) is f x x = ∂ ∂ x ( f x ) . The second order partial derivative of f x with respect to y at the point ( x , y ) is f x y = ∂ ∂ y ( f x ) . The second order partial derivative of f y with respect to y at the point ( x , y ) is f y y = ∂ ∂ y ( f y ) . The second order partial derivative of f y with respect to x at the point ( x , y ) is f y x = ∂ ∂ x ( f y ) . The chain rule is ∂ f ( u ) ∂ x = ∂ f ∂ u ( ∂ u ∂ x ) . Calculation: Differentiate f ( x , y ) = ( 2 x 2 + 3 y 2 ) 3 with respect to x where y is treated as a constant to obtain the first partial derivative of f with respect to x . Use chain rule and simplify as follows. f x = ∂ f ∂ x = ∂ ∂ x ( ( 2 x 2 + 3 y 2 ) 3 ) = 3 ( 2 x 2 + 3 y 2 ) 3 − 1 ( ∂ ∂ x ( 2 x 2 + 3 y 2 ) ) = 3 ( 2 x 2 + 3 y 2 ) 2 ( 4 x ) = 12 x ( 2 x 2 + 3 y 2 ) 2 Differentiate f ( x , y ) = ( 2 x 2 + 3 y 2 ) 3 with respect to y where x is treated as a constant to obtain the first partial derivative of f with respect to y . Use chain rule and simplify as follows. f y = ∂ f ∂ y = ∂ ∂ y ( ( 2 x 2 + 3 y 2 ) 3 ) = 3 ( 2 x 2 + 3 y 2 ) 2 ( ∂ ∂ y ( 2 x 2 + 3 y 2 ) ) = 3 ( 2 x 2 + 3 y 2 ) 2 ( 6 y ) = 18 y ( 2 x 2 + 3 y 2 ) 2 Differentiate each of the functions f x with respect to x to obtain the second order partial derivative f x x , f x x = ∂ ∂ x ( 12 x ( 2 x 2 + 3 y 2 ) 2

Applied Calculus for the Managerial, Life, and Social Sciences

10th Edition

ISBN: 9781305887831

Author: Tan

Publisher: CENGAGE CO

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Chapter 8, Problem 24RE

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To compute: The second order partial derivatives of the function f(x,y)=(2x2+3y2)3 .

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Chapter 8, Problem 23RE

Chapter 8, Problem 25RE

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Chapter 8 Solutions

Applied Calculus for the Managerial, Life, and Social Sciences

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The second order partial derivatives of the function f ( x , y ) = ( 2 x 2 + 3 y 2 ) 3 .

Solution Summary: The author explains the second order partial derivatives of the function f(x,y).

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To compute: The second order partial derivatives of the function f(x,y)=(2x2+3y2)3 .

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Chapter 8 Solutions