Why is the Leibniz rule a sufficient ingredient in the construction of the tangent space?

This is a very soft question, but I am wondering if anyone can shed light on why it is that the product rule (and linearity) provide exactly the right requirement for the space of derivations to be the tangent space? A similar dependence on the product rule seems to pop up if you want to define the cotangent space at p as $m_p / m_p^2$ in the stalk at p of the sheaf of differentiable functions.

What is it that makes the product rule the identifying feature of differentiation among all linear functions?

Solution 1:

I guess this basically boils down to (a simple form of) Taylor's theorem. If you have a linear map $D$ which satisfies the product rule, then it vanishing on constants, since $D(1)=D(1\cdot 1)=D(1)\cdot 1+1\cdot D(1)=2D(1)$. Moreover, if $f$ and $g$ are two functions vanishing in a point $x$, the $D(fg)=D(f)g+fD(g)$ vanishes at $x$. Now the simple form of Taylor's Theorem I was alluding to is that you can write any function $f$ around a point $x$ as a constant plus a linear map acting on $(y-x)$ plus something that vanishes to second order at $x$ and hence can be written as a product of two functions vanishing in $x$. Hence you see that $D(f)$ depends only on the linear part in the Taylor-development, which is exactly what you want from a directional derivative.