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, $u_{xy}=u_{yx}$
