  Treating differentials as independent algebraic units have a long history of
use and abuse. It is generally considered problematic to treat the derivative
as a fraction of differentials rather than as a holistic unit acting as a
limit, though for practical reasons it is often done for the first derivative.
However, using a revised notation for the second and higher derivatives will
allow for the ability to treat differentials as independent units for a much
larger number of cases.
