   Chapter 14.3, Problem 61E

Chapter
Section
Textbook Problem

Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx.61. u = cos(x2y)

To determine

To verify: The conclusion of Clairaut’s Theorem that is uxy=uyx for the function u=cos(x2y) .

Explanation

The given function is, u=cos(x2y) .

Differentiate the given function with respect to x and obtain ux .

ux=sin(x2y)(2xy)

ux=2xysin(x2y) (1)

Differentiate equation (1) with respect to y and obtain the partial derivative, uxy .

2uxy=2x[sin(x2y)(1)+ycos(x2y)(x2)]=2xsin(x2y)2x3ycos(x2y)=2x[sin(x2y)+x2ycos(x2y)]

Hence, uxy=2x[sin(x2y)+x2ycos(x2y)] .

Differentiate the given function with respect to y and obtain uy .

uy=sin(x2y)(x2)

uy=x2sin(x2y) (2)

Differentiate equation (2) with respect to x and obtain the partial derivative, uyx

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