TITLE: How do we take second order of total differential?
QUESTION [7 upvotes]: 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 $d^2f$?

REPLY [6 votes]: $d=dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}$ is a differential operator that when applied to $f$ gives $df=dx\frac{\partial f}{\partial x}+dy\frac{\partial f}{\partial y}$
$d^2=\left(dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}\right)\left(dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}\right)$
$d^2f=\left(dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}\right)\left(dx\frac{\partial}{\partial x}+dy\frac{\partial}{\partial y}\right)f=\left(dx^2\frac{\partial^2}{\partial x^2}+dy^2\frac{\partial^2}{\partial y^2}+2dx dy\frac{\partial^2}{\partial x\partial y}\right)f=dx^2\frac{\partial^2 f}{\partial x^2}+dy^2\frac{\partial^2 f}{\partial y^2}+2dx dy\frac{\partial^2 f}{\partial x\partial y}$