Partial derivatives

a derivative of a function of two or more variables with respect to one variable, the other(s) being treated as constant.

$$\begin{align*} z &= \arctan{\frac{y}{x}} \\ \\ \\ \frac{\partial z}{\partial x} &= \frac{1}{1 + {\frac{y}{x}}^2} \cdot [y \cdot (-1 \cdot x^{-2})] = - \frac{y}{x \left(x + y^{2}\right)} \\ \\ \frac{\partial z}{\partial y} &= \frac{1}{1 + {\frac{y}{x}}^2} \cdot [\frac{1}{x} \cdot 1] = \frac{1}{x + y^2} \\ \\ \\ \frac{\partial^2 z}{\partial x^2} &= \frac{y}{x \left(x + y^{2}\right)^{2}} + \frac{y}{x^{2} \left(x + y^{2}\right)} \\ \\ \frac{\partial^2 z}{\partial y^2} &= - \frac{2 y}{\left(x + y^{2}\right)^{2}} \\ \\ \frac{\partial^2 z}{\partial y \partial x} &= \frac{\partial^2 z}{\partial x \partial y} = - \frac{1}{\left(x + y^{2}\right)^{2}} \end{align*}$$

So, I really wanna say, all those things is about who you should treat as variable, who you should treat as constant

For example:

$\frac{\partial z}{\partial x}$, you should see $x$ as variable, $y$ as constant

$\frac{\partial z}{\partial y}$, you should see $y$ as variable, $x$ as constant

And for $\frac{\partial^2 z}{\partial x \partial y}$, first see $x$ as variable, do calculation, then see $y$ as variable, get result