I have never thought that differentials could be manipulated as numbers. I do not see why that is, in any case, justified.
Can someone prove that the product of dy/du and 1/(dx/du) is equal to dy/dx ? I have seen that if you manipulate the differentials you will get the result easily, but is that justified and what are the domain restrictions in any case that it is justified?
Sorry if my question isn't well phrased.