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?)
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$\begingroup$$\def\part#1#2{{\partial#1\over\partial#2}}$ $\def\parts#1#2#3{{\partial^2#1\over\partial#2\,\partial#3}}$
On the left hand side of your equations, you have the symbol "$\part{\vphantom f}y\bigl(\part f x\bigr)"$. By definition this is the partial derivative of the function $\part fx$ with respect to $y$. So, upon encountering this symbol, you take the function $\part fx$ and then take its partial with respect to $y$. The natural notation of the type on the right hand side of your equations is the notation used in (2) of your post: $$\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.)
$\endgroup$ 2 $\begingroup$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 under the "Clairaut's theorem" subtitle.
$\endgroup$ 11 $\begingroup$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.
$\endgroup$ $\begingroup$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
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