Explanation Given: The function f ( x , y ) = x 2 y + e x y . Calculation: In ordered to obtain the second order partial derivative of f ( x , y ) , first find the first order partial derivative of f ( x , y ) . The first order partial derivative of f ( x , y ) with respect to x is obtained below. f x ( x , y ) = ∂ ∂ x ( x 2 y + e x y ) = y ∂ ∂ x ( x 2 ) + ∂ ∂ x ( e x y ) = y ⋅ 2 x + e x y ⋅ y = 2 x y + y e x y Hence, the first order partial derivative with respect to x is f x ( x , y ) = 2 x y + y e x y . The first order partial derivative of f ( x , y ) with respect to y is obtained below. f y ( x , y ) = ∂ ∂ y ( x 2 y + e x y ) = x 2 ∂ ∂ x ( y ) + ∂ ∂ x ( e x y ) = x 2 ⋅ 1 + e x y ⋅ x = x 2 + x e x y Hence, the first order partial derivative with respect to y is f y ( x , y ) = x 2 + x e x y . Obtain the second order partial derivatives as follows, Differentiate the first order partial derivative f x ( x , y ) = 2 x y + y e x y with respect to x , f x x ( x , y ) = ∂ ∂ x ( 2 x y + y e x y ) = y ∂ ∂ x ( 2 x ) + y ∂ ∂ x ( e x y ) = y ⋅ 2 + y e x y ⋅ y = 2 y + y 2 e x y That is, f x x ( x , y ) = 2 y + y 2 e x y . Differentiate the first order partial derivative f x ( x , y ) = 2 x y + y e x y with respect to y , f x y ( x , y ) = ∂ ∂ y ( 2 x y + y e x y ) = 2 x ∂ ∂ y ( y ) + ∂ ∂ y ( y e x y ) = 2 x ⋅ 1 + y

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, Problem 2BM

To determine

To find: The first and second partial derivatives of the function f(x,y)=x2y+exy .

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Chapter 8, Problem 1BM

Chapter 8, Problem 3BM

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Chapter 8 Solutions

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

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Knowledge Booster

The first and second partial derivatives of the function f ( x , y ) = x 2 y + e x y .

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

To find: The first and second partial derivatives of the function f(x,y)=x2y+exy .

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Chapter 8 Solutions