Partial Differentiation: second partial derivative

I am not quite sure how [tex]\frac{\partial}{\partial u}\left(\frac{\partial z}{\partial u}\right)[/tex]

[tex]=\frac{\partial}{\partial u}\left( u \frac{\partial z}{\partial x}+v\frac{\partial z}{\partial y} \right)[/tex]



comes to [tex]\frac{\partial z}{\partial x} + u\frac{\partial}{\partial u}\left(\frac{\partial z}{\partial x}\right)+v \frac{\partial}{\partial u}\left(\frac{\partial z}{\partial y}\right)[/tex]...





I would have evaluated this to be the same but without the [itex]\frac{\partial z}{\partial x}[/itex] term. i.e.:



[tex]u\frac{\partial}{\partial u}\left(\frac{\partial z}{\partial x}\right)+v \frac{\partial}{\partial u}\left(\frac{\partial z}{\partial y}\right)[/tex]



I know I am probably overlooking something obvious/simple but I can't see what it is. Could anyone tell me?