I have read from multiple sources that dy/dx is not to be interpreted as a ratio as the idea of 'dy' and 'dx' themselves will lead to logical difficulties.
However, I have seen in many areas (e.g. thermodynamics) where notations involving dx etc. are used constantly. For example, in http://en.wikipedia.org/wiki/Total_differential, it seems to tread df/dt as a ratio by multiplying dt on both sides to cancel out the denominator.
So in what cases are we allowed to manipulate the notation as if they are quotients?