$\sqrt{1 + (\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2} \, dx \, dy$

$(\frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}) \, dx \, dy$

$\sqrt{(\frac{\partial z}{\partial x})^2 + (\frac{\partial z}{\partial y})^2} \, dx \, dy$

$(1 + \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y}) \, dx \, dy$