  Differential operators usually result in derivatives expressed as a ratio of
differentials. For all but the simplest derivatives, these ratios are typically
not algebraically manipulable, but must be held together as a unit in order to
prevent contradictions. However, this is primarily a notational and conceptual
problem. The work of Abraham Robinson has shown that there is nothing
contradictory about the concept of an infinitesimal differential operating in
isolation. In order to make this system extend to all of calculus, however,
some tweaks to standard calculus notation are required. Understanding
differentials in this way actually provides a more straightforward
understanding of all of calculus for students, and minimizes the number of
specialized theorems students need to remember, since all terms can be freely
manipulated algebraically.
