Explanation Given information: The function is f ( x , y ) = 2 x y . Formula used: For a function, f ( x , y ) = f ( y ) ⋅ f ( x ) the partial derivative with respect to x is calculated as, ∂ ∂ x [ u ( x ) ] = ∂ ∂ x [ f ( y ) ⋅ f ( x ) ] = [ f ( y ) ] ∂ ∂ x [ f ( x ) ] 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 function. f ( x , y ) = 2 x y Use partial derivative with respect to x for the calculation of f x . f x = ∂ ∂ x ( 2 x y ) = 2 y ( ∂ x ∂ x ) = ( 2 y ) ⋅ 1 = 2 y Rewrite the expression for f x

Calculus and Its Applications (11th Edition)

11th Edition

ISBN: 9780321979391

Author: Marvin L. Bittinger, David J. Ellenbogen, Scott J. Surgent

Publisher: PEARSON

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Chapter 6.2, Problem 27E

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To calculate: The four second order-partial derivatives for the function f(x,y)=2xy.

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Chapter 6.2, Problem 26E

Chapter 6.2, Problem 28E

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