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).

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"!) Also note that the chain-rule is more or less encoded in the definition (easy to see via charts and atlases).

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).

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.