As you pointed out infintessimal manipulation in physics produces the correct results most of the time. In fact, it's almost always correct because Taylor Serie

As you pointed out infintessimal manipulation in physics produces the correct results most of the time. In fact, it's almost always correct because Taylor Series say so. In the physical world we deal with differentials that are always real. Technically physicists are rejecting the later terms dealing with convergence because for all practical purposes they don't matter. This frustrates math people like me to no end but I can see why they do that. To a mathematician dx is certainly not a number. It's an infintessimal and therefore not a real number. But physicists can get away with it...their math is simpler.