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I am reading the thread but I can't comment. So I would like to open a new one and ask for clarification. Could you explain the difference between "vector" and "point" here?

His is the answer of John Robertson:

Of course it is a ratio. dy and dx are differentials. Thus they act on tangent vectors, not on points. That is, they are functions on the tangent manifold that are linear on each fiber. On the tangent manifold the ratio of the two differentials dy/dx is just a ratio of two functions and is constant on every fiber (except being ill defined on the zero section) Therefore it descends to a well defined function on the base manifold. We refer to that function as the derivative.

As pointed out in the original question many calculus one books these days even try to define differentials loosely and at least informally point out that for differentials dy=f′(x)dxdy=f′(x)dx (Note that both sides of this equation act on vectors, not on points). Both dy and dx are perfectly well defined functions on vectors and their ratio is therefore a perfectly meaningful function on vectors. Since it is constant on fibers (minus the zero section), then that well defined ratio descends to a function on the original space.

At worst one could object that the ratio dy/dx is not defined on the zero section.

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The generalisation of calculus from vector spaces to manifolds relies on a crucial observation, namely that elements of vector spaces have two properties at once: they are both a "vector" and a "point". So what is meant by these words? Loosely speaking, a vector stands for a direction whereas a point indicates a location.

You can see this in the following example. Consider a (smooth) function between (finite-dimensional) vector spaces $f:V\rightarrow W$. Denote its derivative by $Df$. Then the directional derivative of $f$ in direction $v\in V$ is defined as $$ D_vf: V\rightarrow W, \quad p\mapsto D_vf(p):=Df_p(v)\equiv Df(p)(v) $$ So $v\in V$ denotes the direction in which we want to differentiate, whereas $p\in V$ is the point where we want to evaluate the derivative.

A first abstraction is done by generalising this concept to affine spaces. An affine space is a tuple $(A,\vec A)$ consisting of a set $A$ (the "points"), a vector space $\vec A$ (obviously the "vectors") and a mapping $+: A\times \vec A \rightarrow A, (p,v)\mapsto p + v$ ("the translations"), which works exactly as in vector spaces (for details see Wikipedia).

Then the derivative of a function $f$ between affine spaces $A$ and $B$ is defined as in vector spaces (we just have to replace addition with translation). It now represents a map $$ Df: A\rightarrow \operatorname{Hom}(\vec A, \vec B), \quad p \mapsto Df_p, $$ i.e. $Df_p$ is a linear map between vectors in $\vec A$ and vectors in $\vec B$.

Especially, if $B=\mathbb R$, then at each point $Df=df$ is a linear functional on $\vec A$ which makes $df$ a differential 1-form on $A$. In differential geometry, differentials are thus seen (equivalently!) as mappings from vectors to functions $$ df: \vec A \rightarrow C^\infty(A), \quad v \mapsto df(v). $$

At this point you probably already understood that there is a difference between points and vectors on the level of affine/vector spaces. I will now say some words on how these concepts are carried over to manifolds by differential geometry. However, I will only present some ideas. If you are interested, you should attend either a lecture or have a look in a book (e.g. J. M. Lee, "Introduction to Smooth Manifolds").

Another point of view is to think of the vectors $v\in\vec A$ as to be attached to points $p \in A$, formally we think of the product $A \times \vec A$. Then a pair $(p,v)$ can act on smooth functions by the directional derivative: $$ (p,v): C^\infty(A) \rightarrow \mathbb R, \quad f \mapsto df_p(v) $$ The idea of differential geometry is to take this point of view and make this concept local. I.e. allow the directional space $\vec A$ to vary from point to point. More precisely, the above concept is even taken as the definition: The directional space at $p$, the tangent space, is defined to consist of all such functionals $(p,v)$.

Note that the formalisation of this idea is a lot more work since you first have to define manifolds, declare a smooth structure (to know what a smooth function is) and then construct the tangent spaces based on the idea above.

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