Partial Derivatives Notation

Suppose that we have the function $$f : \mathbb{R}^2 \rightarrow \mathbb{R}$$ which is a function of two real variables. 

We denote its second order partial derivatives in the following way

$$\frac{\partial^2 f}{\partial x^2}$$

$$\frac{\partial^2 f}{\partial y^2}$$

$$\frac{\partial^2 f}{\partial x \partial y}$$

$$\frac{\partial^2 f}{\partial y \partial x}$$

In this notation it is important to remember that by definition

$$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial f}{\partial x}\left(\frac{\partial f}{\partial y}\right)$$

$$\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial f}{\partial y}\left(\frac{\partial f}{\partial x}\right)$$

So if we write x first in the denominator, that means x is the second variable we differentiate by, 

and if we write y first in the denominator, that means y is the second variable we differentiate by. 

In other words, first we differentiate by the right-most variable (from the denominator), second we differentiate by the second right-most variable (from the denominator) and so on. Of course these are not really denominators (in the algebraic sense) but OK, they look like denominators.

The same applies to functions of more than two variables.

Suppose $$g : \mathbb{R}^3 \rightarrow \mathbb{R}$$ is a function of three real variables. 

Then by definition the expression 

$$\frac{\partial^3 g}{\partial x \partial y \partial z}$$ means the following

$$\frac{\partial^3 g}{\partial x \partial y \partial z} = \frac{\partial g}{\partial x}\left(\frac{\partial g}{\partial y} \left(\frac{\partial g}{\partial z}\right)\right)$$

In other words here we have differentiated first by $z$, then by $y$, and finally by $x$.