## Calculus 8th Edition

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