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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.

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  • $\begingroup$ To me a derivative is the function without memory :p $\endgroup$ – Masacroso Feb 18 '16 at 17:41
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    $\begingroup$ You would need to know what a connection is and what a Lagrangian submanifold is. For the first, read any book on differential geometry that talks about connections. I like Taubes (but that assumes a first manifolds course, eg Lee). Then you want to read the beginning of a symplectic manifolds book. Keep in mind that this definition is mostly a joke; I would be astonished if there was a mathematician who actually found this an inspiring definition of the derivative. See also this answer. $\endgroup$ – user98602 Feb 18 '16 at 17:42

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