Explanation Given information: The equation is f ( x , y ) = x 5 y 4 + x 3 y 2 . Formula used: For a function z = f ( x , y ) , For ∂ z ∂ x = f x , x is treated as variable and y is treated as a constant and the function is differentiated. For ∂ 2 z ∂ x 2 = ∂ f x ∂ x = f x x , x is treated as variable and y is treated as a constant and the function is double differentiated with respect to x . For ∂ z ∂ y = f y , y is treated as variable and x is treated as a constant and the function is differentiated. For ∂ 2 z ∂ y 2 = ∂ f y ∂ y = f y y , y is treated as variable and x is treated as a constant and the function is double differentiated with respect to y . For ∂ 2 z ∂ x ∂ y = ∂ f y ∂ x = f y x , y is treated as variable and x is treated as a constant for f y and x is treated as variable and y is treated as a constant for ∂ f y ∂ x . For ∂ 2 z ∂ y ∂ x = ∂ f x ∂ y = f x y , x is treated as variable and y is treated as a constant for f x and y is treated as variable and x is treated as a constant for ∂ f x ∂ y . Calculation: Consider the equation. f ( x , y ) = x 5 y 4 + x 3 y 2 Use the partial derivative with respect to x for the calculation of f x . f x = ∂ ∂ x ( x 5 y 4 + x 3 y 2 ) = y 4 ∂ ∂ x x 5 + y 2 ∂ ∂ x x 3 = y 4 ( 5 x 4 ) + y 2 ( 3 x 2 ) = 5 x 4 y 4 + 3 x 2 y 2 Rewrite the expression for f x . f x = 5 x 4 y 4 + 3 x 2 y 2 …… (1) Apply the partial derivative with respect to x on f x for calculation of f x x . f x x = ∂ ∂ x ( 5 x 4 y 4 + 3 x 2 y 2 ) = 5 y 4 ∂ ∂ x x 4 + 3 y 2 ∂ ∂ x x 2 = 5 y 4 ( 4 x 3 ) + 3 y 2 ( 2 x ) = 20 x 3 y 4 + 6 x y 2 Take the partial derivative of equation (1) with respect to y for the calculation of f x y

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ISBN: 9781323161470

Author: BITTINGER

Publisher: PEARSON C

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Chapter 6.2, Problem 31E

To determine

To calculate: The four second order-partial derivatives for the function f(x,y)=x5y4+x3y2.

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Chapter 6.2, Problem 30E

Chapter 6.2, Problem 32E

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The four second order-partial derivatives for the function f ( x , y ) = x 5 y 4 + x 3 y 2 .

Solution Summary: The author calculates the four second-order-partial derivatives for the function f(x,y).

To calculate: The four second order-partial derivatives for the function f(x,y)=x5y4+x3y2.

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