Answer
$\displaystyle \frac{\partial z}{\partial x}=y\sec^{2}xy,\quad \displaystyle \frac{\partial z}{\partial y}=x\sec^{2}xy$
Work Step by Step
Treat y as constant to calculate$ \displaystyle \frac{\partial z}{\partial x}$
$\displaystyle \frac{\partial z}{\partial x}\stackrel{\text{chain rule} }{=}(\sec^{2}xy)[\frac{\partial}{\partial x}(xy)]=(\sec^{2}xy)(y)=y\sec^{2}xy$
Treat x as constant to calculate$\displaystyle \frac{\partial z}{\partial y}$
$\displaystyle \frac{\partial z}{\partial y}\stackrel{\text{chain rule} }{=}(\sec^{2}xy)(x)=x\sec^{2}xy$
