Explanation Given: The function is z = x 2 + y 2 . Formula Used: The partial derivative of the function is given as: f x ( x , y ) = ∂ ∂ x ( f ( x , y ) ) The derivative formula for the function x n : d d x x n = n x n − 1 Calculation: Consider the function: z = x 2 + y 2 …… (1) Calculate the partial derivative of the function with respect to x assuming y constant as follows: z x = ∂ ∂ x ( x 2 + y 2 ) = 1 2 x 2 + y 2 ∂ ∂ x ( x 2 + y 2 ) = 1 2 x 2 + y 2 ( 2 x ) Simplify the above equation, z x = x x 2 + y 2 …… (2) Now, calculate the partial derivative of the above equation with respect to x again: z x x = ∂ ∂ x ( x x 2 + y 2 ) = x 2 + y 2 ∂ ∂ x x − x ∂ ∂ x x 2 + y 2 ( x 2 + y 2 ) 2 = x 2 + y 2 − x 2 x 2 x 2 + y 2 ( x 2 + y 2 ) 2 = ( x 2 + y 2 ) − x 2 ( x 2 + y 2 ) 3 / 2 Simplify this further and get: z x x = y 2 ( x 2 + y 2 ) 3 / 2 Similarly, calculate the partial derivative of the function with respect to y assuming x constant as follows: z y = ∂ ∂ y ( x 2 + y 2 ) = 1 2 x 2 + y 2 ∂ ∂ y ( x 2 + y 2 ) = 1 2 x 2 + y 2 ( 2 y ) = y x 2 + y 2 …… (3) Now, calculate the partial derivative of the above equation with respect to y again: z y y = ∂ ∂ y ( y x 2 + y 2 ) = x 2 +

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ISBN: 9781285057095

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 13.3, Problem 77E

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To calculate: The four second partial derivative of function z=x2+y2.

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Chapter 13.3, Problem 76E

Chapter 13.3, Problem 78E

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The four second partial derivative of function z = x 2 + y 2 .

Solution Summary: The author explains the four second partial derivative of the function z=sqrtx2+y2.

To calculate: The four second partial derivative of function z=x2+y2.

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Chapter 13 Solutions