Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 14 - Partial Derivatives - 14.3 Partial Derivatives - 14.3 Exercises: 59

Answer

$u_{xy}=12x^{3}y^{2}$ and $u_{yx}=12x^{3}y^{2}$ Hence, $u_{xy}=u_{yx}$

Work Step by Step

Consider the function $u=x^{4}y^{3}-y^{4}$ Need to prove the conclusion of Clairaut’s Theorem holds, that is, $u_{xy}=u_{yx}$ In order to find this differentiate the function with respect to $x$ keeping $y$ constant. $u_{x}=4x^{3}y^{3}$ Differentiate $u_{x}$ with respect to $y$ keeping $x$ constant. $u_{xy}=12x^{3}y^{2}$ Differentiate the function with respect to $y$ keeping $x$ constant. $u_{y}=3x^{4}y^{2}-4y^{3}$ Differentiate $u_{y}$ with respect to $x$ keeping $y$ constant. $u_{yx}=12x^{3}y^{2}$ Hence, $u_{xy}=u_{yx}$
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