What does it mean to extend the manipulability of differentials?

In an interesting paper called Extending the Manipulability of Differentials, the authors Jonathan Bartlett and Asatur Zh. Khurshudyan describe an interesting proposal for representing higher-order derivatives. The argument is basically this:

    As is well-known, the chain rule for first derivatives seems to follow algebraically if you use Leibniz notation for the derivatives: (dg/df)(df/dx) = dg/dx
    However, there’s a chain rule for second derivatives and it doesn’t seem to follow algebraically even when using the Leibniz notation d^2f/dx^2 for the second derivative
    But (the authors argue), it does if we use a different notation for the second derivative: d(df/dx)/dx, and furthermore this can be rewritten by expanding d(df/dx) using the quotient rule

This is quite intriguing. But I found it a bit odd that the authors presented this as a purely notational idea, when there seems to be some non-trivial semantic thing going on here. This is one attempt at spelling out what it is:

Let S be the free smooth algebra with generators x_1, x_2, \ldots, dx_1, dx_2, \ldots, d^2x_1, d^2x_2, \ldots (right now, these are just names and the d has no meaning). Then the proposition is:

There exists a function d\colon S\to S such that

  • d(d^nx_i)=d^{n+1}x_i (including the case d^0x_i=x_i)
  • For all smooth functions F\colon \mathbb{R}^n\to\mathbb{R} and e_1,\ldots,e_n\in S, d(F(e_1,\ldots,e_n)=F_1(e_1,\ldots,e_n)de_1 + \cdots + F_n(e_1,\ldots,e_n)de_n, where F_i is the ith partial derivative of F.

This is non-trivial (and thus useful) because a given element e\in S might be expressible in more than one way by the rules above.

This is likely not a complete characterization of all the ways you could use the rules in the paper. In particular, this only deals with total smooth functions, so even an expression like e_1/e_2 isn’t directly dealt with (you have to say that e_1/e_2 means an e_3 such that e_2 e_3 = e_1), but I think it captures a good bit of what’s going on.