\(u = x^y\)
\(\frac{\partial u}{\partial x} = yx^{y-1}\)
\(\frac{\partial^2 u}{\partial x^2} = y(y-1)x^{y-2}\)
\(\frac{\partial^3 u}{\partial x^2\partial y} = (y^2 - y) x^{y-2} log x + (2y - 1)x^{y-2}\)
\(\frac{\partial ^2u}{\partial x\partial y} = yx^{y-1}logx + x^{y-1}\)
\(\frac{\partial ^2u}{\partial x\partial y \partial x} = y(y-1) x^{y-2} logx + \frac{yx^{y-1}}{x} + (y -1)x^{y-2}\)
\(= x^{y-2} (y^2 - y) logx + x^{y-2} (2y - 1)\)
\(\frac{\partial ^3u}{\partial x^2\partial y} = \frac{\partial ^2u}{\partial x\partial y \partial x} \)
Hence proved.
