why $\frac{\partial}{\partial r}(\frac{\partial z}{\partial x})$ is not $\frac{\partial}{\partial x}(\frac{\partial z}{\partial x})\frac{\partial x}{\partial r}$ but $\frac{\partial}{\partial x}(\frac{\partial z}{\partial x})\frac{\partial x}{\partial r}+\frac{\partial}{\partial y}(\frac{\partial z}{\partial x})\frac{\partial y}{\partial r}$?
Since $z=f(x,y)$, we have to assume, for generality, that $\partial z / \partial x$ is also a function of both $x$ and $y$ (this would not be true only if any $y$ disappears in the derivation). So we have $\partial z_x / \partial r$, where $z_x = z_x(x,y)$. The chain rule has to contemplate both variables. Just imagine, for example, if it had to contemplate only one variable. It wouldn't make sense to choose $x$ over $y$, or vice-versa.
You should check out this link here for some more informatino and explanation.