A brief note about derivatives
A blog post led me to a paper, “Extending the Algebraic Manipulability of Differentials”, which makes a useful point about the notation we use for derivatives. This is a brief summary so I don’t forget it.^{1}
Observation: the derivative operator can be decomposed into two steps: applying the differential operator to the target, then dividing by . It is useful to think of this as occuring in two steps, because it removes confusion in certain notations. Particularly, we will identify these two ways of writing the second derivative as meaning slightly different things:^{2}
vs

The paper itself has a bit of a sketchy pseudoacademic quality to it, spending a lot of time explaining things that every mathematician should know – but a good point is a good point, and I like any effort to improve notation. ↩

note that this is not the same use of as is used in exterior algebra, with . That one requires additionally quotienting by relations like . ↩