Explanation Given: The given function is f ( x , y ) = − 7 x 2 x + 3 y . Approach: To find f x ( x , y ) differentiate the f ( x , y ) with respect to x by holding y . f x ( x , y ) = ∂ ∂ x f ( x , y ) To find f x x ( x , y ) differentiate the f ( x , y ) with respect to x by holding y . f x x ( x , y ) = ∂ ∂ x f x ( x , y ) = ∂ 2 ∂ x 2 f ( x , y ) To find f y ( x , y ) differentiate the f ( x , y ) with respect to y by holding x . f y ( x , y ) = ∂ ∂ y f ( x , y ) To find f y y ( x , y ) differentiate the f ( x , y ) with respect to x by holding y . f y y ( x , y ) = ∂ ∂ y f y ( x , y ) = ∂ 2 ∂ y 2 f ( x , y ) The quotient rule of partial derivative with respect to y , ∂ ∂ y [ f ( x , y ) h ( x , y ) ] = h ( x , y ) ∂ ∂ y f ( x , y ) − f ( x , y ) ∂ ∂ y h ( x , y ) [ h ( x , y ) ] 2 The quotient rule of partial derivative with respect to x , ∂ ∂ x [ f ( x , y ) h ( x , y ) ] = h ( x , y ) ∂ ∂ x f ( x , y ) − f ( x , y ) ∂ ∂ x h ( x , y ) [ h ( x , y ) ] 2 Calculation: Consider the given function, f ( x , y ) = − 7 x 2 x + 3 y To find f x ( x , y ) differentiate the f ( x , y ) with respect to x by holding y . f x ( x , y ) = ∂ ∂ x ( − 7 x 2 x + 3 y ) Apply the quotient rule of derivative, ∂ ∂ x ( − 7 x 2 x + 3 y ) = − ( 2 x + 3 y ) ∂ ∂ x 7 x − 7 x ∂ ∂ x ( 2 x + 3 y ) ( 2 x + 3 y ) 2 = − ( 2 x + 3 y ) ( 7 ) − 14 x ( 2 x + 3 y ) 2 = − 14 x + 21 y − 14 x ( 2 x + 3 y ) 2 = − 21 y ( 2 x + 3 y ) 2 Thus, f x ( x , y ) = − 21 y ( 2 x + 3 y ) 2 To find second derivate f x x ( x , y ) differentiate the f x ( x , y ) with respect to x by holding y . f x x ( x , y ) = ∂ ∂ x ( − 21 y ( 2 x + 3 y ) 2 ) Apply the quotient rule of derivative, ∂ ∂ x ( − 21 y ( 2 x + 3 y ) 2 ) = − ( 2 x + 3 y ) 2 ∂ ∂ x 21 y − 21 y ∂ ∂ x ( 2 x + 3 y ) 2 ( 2 x + 3 y ) 4 = − ( 2 x + 3 y ) 2 ( 0 ) − 21 y [ 2 ( 2 x + 3 y ) 2 ] ( 2 x + 3 y ) 4 = 84 y ( 2 x + 3 y ) 3 To find r y ( x , y ) differentiate the r ( x , y ) with respect to y by holding x . r y ( x , y ) = ∂ ∂ y ( − 7 x 2 x + 3 y ) Apply the quotient rule of derivative, ∂ ∂ y ( − 7 x 2 x + 3 y ) = − ( 2 x + 3 y ) ∂ ∂ y 7 x − 7 x ∂ ∂ y ( 2 x + 3 y ) ( 2 x + 3 y ) 2 = − ( 2 x + 3 y ) ( 0 ) − 7 x ( 3 ) ( 2 x + 3 y ) 2 = 21 x ( 2 x + 3 y ) 2 Thus, f y ( x , y ) = 21 x ( 2 x + 3 y ) 2 To find second derivate f y y ( x , y ) differentiate the f y ( x , y ) with respect to y by holding x

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

2nd Edition

ISBN: 9780137553457

Author: Raymond Greenwell, Nathan Ritchey

Publisher: PEARSON+

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Question

Chapter 9.2, Problem 28E

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To find:

The second-order partial derivatives for the function k(x,y)=−7x2x+3y.

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Chapter 9.2, Problem 27E

Chapter 9.2, Problem 29E

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Chapter 9 Solutions

Pearson eText for Calculus for the Life Sciences -- Instant Access (Pearson+)

Ch. 9.1 - YOUR TURN For the function in Example 1, find...Ch. 9.1 - Prob. 2YT Ch. 9.1 - Prob. 3YT Ch. 9.1 - EXERCISES Let f(x,y)=2x3y+5.Find the following. a....Ch. 9.1 - Prob. 2E Ch. 9.1 - EXERCISES Let h(x,y)=x2+2y2. Find the following....Ch. 9.1 - EXERCISES Let f(x,y)=9x+5ylogx. Find the...Ch. 9.1 - EXERCISES Let f(x,y)=ex+ln(x+y). Find the...Ch. 9.1 - EXERCISES Let f(x,y)=xex+y. Find the following. a....Ch. 9.1 - EXERCISES Let f(x,y)=xsin(x2y). Find the...

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The second-order partial derivatives for the function k ( x , y ) = − 7 x 2 x + 3 y .

Solution Summary: The author explains the second-order partial derivatives for the function k(x,y)=-7x2x+3y

To find:

The second-order partial derivatives for the function k(x,y)=−7x2x+3y.

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Chapter 9 Solutions

The second-order partial derivatives for the function k ( x , y ) = − 7 x 2 x + 3 y .

Solution Summary: The author explains the second-order partial derivatives for the function k(x,y)=-7x2x+3y

To find: The second-order partial derivatives for the function k(x,y)=−7x2x+3y.

Want to see the full answer?

Chapter 9 Solutions

To find:

The second-order partial derivatives for the function k(x,y)=−7x2x+3y.