How do we take second order of total differential? This is the total differential 
$$df=dx\frac {\partial f}{\partial x}+dy\frac {\partial f}{\partial y}.$$
How do we take higher orders of total differential, $d^2 f=$?
Suppose I have $f(x,y)$ and I want the second order total differential $d^2f$?

Edit: I interpret your $d^2 f$ as $d^2f=d(df)$.
If $f$ is $C^2(\Omega)$, with $(x,y)\in\Omega$, then
$$d(df)=dy\wedge dx \frac{\partial^2 f}{\partial y\partial x}+dx\wedge dy 
\frac{\partial^2 f}{\partial x\partial y}=\left(\frac{\partial^2 f}{\partial x\partial y}- \frac{\partial^2 f}{\partial y\partial x} \right)dx\wedge dy=0,$$
as $dx \wedge dy=-dy\wedge dx$ and $\frac{\partial^2 f}{\partial x\partial y}= \frac{\partial^2 f}{\partial y\partial x}$ on $\Omega$.
If $f$ is not $C^2$ at $(x,y)$, then we arrive at
$$d(df)=\left(\frac{\partial^2 f}{\partial x\partial y}- \frac{\partial^2 f}{\partial y\partial x} \right)dx\wedge dy. $$
