Second order partial derivatives - notation I have seen both of these used, and people around me seem to disagree, so which one is correct: (first derivative with respect to x, then y): 
(1) $$\frac{\partial }{\partial y}(\frac{\partial f}{\partial x}) = \frac{\partial^{2} f}{\partial x\partial y}$$ 
(2) $$\frac{\partial }{\partial y}(\frac{\partial f}{\partial x}) = \frac{\partial^{2} f}{\partial y\partial x}$$
and why? (reasons, history?)
 A: $\def\part#1#2{{\partial#1\over\partial#2}}$
$\def\parts#1#2#3{{\partial^2#1\over\partial#2\,\partial#3}}$
On the left hand side of your equations, you have the symbol "$\part{\vphantom f}y\bigl(\part f x\bigr)"$.
By definition this  is the partial derivative of the function $\part fx$ with respect to $y$. 
So, upon encountering this symbol, you   take the function $\part fx$ and  then take its partial with respect to $y$.  The natural notation of the type on the right hand side of your equations  is the notation used in (2) of your post:
$$\tag{3}
\part{\vphantom f}y\Bigl(\part f x\Bigr)=\parts f y x.
$$
I will not surmise why this is the "natural" notation, but will point out that $(3)$ gives  the adopted definition for the symbol $\parts f y x$ in any calculus/analysis text, or  any other "credible" source, you'll find.
I emphasise here  that $(3)$ defines the symbol $\parts f y x$; that this sometimes gives an expression that equals $\parts f x y$ is irrelevant. (Of course, for certain functions, what you wrote in (1) would be correct; but its correctness would follow from the result of a theorem, not from the definition of the symbols.)
A: The order is important when the function is not $C^2$. That is, the second derivatives (in relation to any combination of two variables) of $f$ are continuous functions. If the function is $C^2$ then it doesn't matter the order in which the variables appear.
This is a widely known result called Schwarz's Theorem, but it seems that there are other names for it. Check out for more in http://en.wikipedia.org/wiki/Symmetry_of_partial_derivatives under the "Clairaut's theorem" subtitle.
A: here i uesd notation $f_{xy}$ and $f_{yx}$ for double derivative.\
There is some theorem on equality of $f_{xy}$ and $f_{yx}$.
Theorem : (Young's theorem)\
Let $f$ be a real function defined on non-empty open set $E$ subset of $R^{2}$. If $f_{x}$ and $f_{y}$ exist in some nbhd. of $(x,y)$ and are both differentiable at point $(x,y)$ with respect to $x$ and $y$ then $f_{xy}=f_{yx}$ at point $(x,y)$.
Theorem: (Schwartz's theorem)
Let $f:E\to R$ be a function such that its partial derivatives $f_{x},f_{y}$ and $f_{xy}$ exist and are continuous in a nbhd. of a point $(x,y)$ then $f_{yx}$ exist such that $f_{xy}=f_{yx}$.
Note: One can see that the converse of above theorem need not be true.
A: I always saw the second : the last operation you made is derivating with respect Y and tha symbol appears in the same order .
In the other notation the order is inverse fxy it means the last operation you made is derivating with respect y.
But : if both fxy and fyx are defined at a neighborhood of a point and are continuous at the point their value is the same at the point.See :Mathematical Analysis , Tom .A. Apostol, Definition, 6-10
