PX275 - A2b - testing for exact differentials

for a function, $f (x, y)$ , the total differential is: $d f = (\frac{\partial f}{\partial x}) d x + (\frac{\partial f}{\partial y}) d y$
writing $\frac{\partial f}{\partial x}$ as $A$ , and $\frac{\partial f}{\partial y}$ as $B :$ $d f = A d x + B d y$
now, differentiating $A$ and $B$ wrt $y$ and $x$ respectively:

\begin{matrix} \frac{\partial A}{\partial y} = \frac{\partial}{\partial y} (\frac{\partial f}{\partial x}) = \frac{\partial^{2} f}{\partial y \partial x} \\ \frac{\partial B}{\partial x} = \frac{\partial}{\partial x} (\frac{\partial f}{\partial y}) = \frac{\partial^{2} f}{\partial x \partial y} \\ ∴ \frac{\partial A}{\partial y} = \frac{\partial B}{\partial x} \end{matrix}

if $\frac{\partial A}{\partial y} \neq \frac{\partial B}{\partial x}$ , it is an inexact differential, ie: $f (x, y)$ does not exist