In the paper On proof and progress in mathematics, W. P. Thurston gives the following interpretation of the derivative.
...one person’s clear mental image is another person’s intimidation:
The derivative of a real-valued function $f$ in a domain $D$ is the Lagrangian section of the cotangent bundle $T^∗(D)$ that gives the connection form for the unique flat connection on the trivial $\mathbf R$-bundle $D\times \mathbf R$ for which the graph of $f$ is parallel.
In order to make this interpretation clear, what should a person who only finished a "traditional" Calculus course (say in the James Stewart's style) read?
Indications of books or chapter of books would be great.