Answer
$\frac{\partial^{2} w}{\partial x^{2}}$ = $2x^{2}y^{2}[sec^{2}(xy)tan(xy)]+4xy[sec^{2}(xy)]+2tan(xy)$
$\frac{\partial^{2} w}{\partial y^{2}}$ = $2x^{4}[sec^{2}(xy)tan(xy)]$
$\frac{\partial^{2} w}{\partial x {\partial y}}$ = $2x^{3}y[sec^{2}(xy)tan(xy)]+3x^{2}[sec^{2}(xy)]$
Work Step by Step
$\frac{\partial w}{\partial x}$ = $x^{2}sec^{2}(xy)[y]+tan(xy)[2x]$ = $x^{2}y[sec^{2}(xy)]+2x[tan(xy)]$
$\frac{\partial^{2} w}{\partial x^{2}}$ = $x^{2}y(2)[sec^{2}(xy)tan(xy)][y]+[sec^{2}(xy)](2xy)+(2x)sec^{2}(xy)[y]+2tan(xy)$ = $2x^{2}y^{2}[sec^{2}(xy)tan(xy)]+4xy[sec^{2}(xy)]+2tan(xy)$
$\frac{\partial w}{\partial y}$ = $x^{2}sec^{2}(xy)[x]$ = $x^{3}[sec^{2}(xy)]$
$\frac{\partial^{2} w}{\partial y^{2}}$ = $x^{3}[2sec^{2}(xy)tan(xy)][x]$ = $2x^{4}[sec^{2}(xy)tan(xy)]$
$\frac{\partial^{2} w}{\partial x {\partial y}}$ = $x^{2}y(2)[sec^{2}(xy)tan(xy)][x]+[sec^{2}(xy)](x^{2})+(2x)sec^{2}(xy)[x]$ = $2x^{3}y[sec^{2}(xy)tan(xy)]+3x^{2}[sec^{2}(xy)]$
