Todd Trimble's answer that the derivative in the disguise of the tangent mapping is a functor is the perfect answer. But I want to elaborate a little also on his first answer:
- What are elementary functions?
An answer that I often gave in introductory courses is: Solutions of linear ODEs with specal constant (integer?) coefficients of low order.
If we would give names to solutions of corresponding integral equations, maybe then formal integration would look easier, as it is numerically.
Answer to the comment:
From elementary functions (maybe better, functions with names) we construct other functions by composition, etc. These we want to integrate.
For formal integration we could consider expressions like $df = f'dx$ with the usual rules; in particular $d(f\circ g) = (f'\circ g) g' dx$ and integration by parts $d(fg) = f.dg + g.df$ Call anything of the form $df$ a total differential. Then the game is: Reduce it to a sum of total differentials. Then remove all $d$'s, and you integrated the given 1-form $fdx$. Let me paraphrase this as: Formal integration is a cohomological operation (or anti-operation).
