Answer
$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=xy+z$
Work Step by Step
Given: $z=xy+e^{y/x}$
$\frac{\partial z}{\partial x}=y+e^{y/x}-\frac{ye^{y/x}}{x}$
$\frac{\partial z}{\partial y}=x+e^{y/x}$
$x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=x(y+e^{y/x}-\frac{ye^{y/x}}{x})+y(x+e^{y/x})$
Hence, $x\frac{\partial z}{\partial x}+y\frac{\partial z}{\partial y}=xy+z$
