I just want to quickly mention a connection (sorry about the pun) that hasn't been mentioned yet: the product rule may be thought of as being *equivalent* to integration by parts which in turn allows us extend the notion of differentiation to larger function spaces (e.g., in analysis of PDEs and the theory of distributions therein; see: [distribution][1]).

I should also note that defining the tangent space / differentiation via curves is equivalent to defining them via charts and atlases.

Finally, linearity + Leibniz's (product) rule yield all the basic properties we require for differentiation (and actually there aren't any other properties that are "left over"!). See Ben Crowell's comment and links therein.

  [1]: http://en.wikipedia.org/wiki/Distribution