Motivation for the definition of a differentiable structure on a manifold

Question

I'm writing a report on differentiable manifolds and while I know that using an atlas of smooth charts gives a incredibly useful definition of what it means for a function on a manifold M to be differentiable, I'm wondering about the actual need for adding such extra structure. Lots of authors (e.g. Spivak) say that on a general topological manifold M there isn't a meaningful definition of the derivative of a real-valued function on M. However the definition of a manifold usually requires it to be second countable Hausdorff and so the manifold is metrisable giving a meaning for the distance between points. So why not define the derivative of a function $f:M\to\mathbb{R}$ at a point $x \in M$ by the limit of $\frac{f(x) - f(y)}{d(x,y)}$ as y approaches x?

Answer 1

1. The value of that limit generally depends on the "direction" in which $y$ approaches $x$ as soon as the dimension of $M$ is bigger than $1$. The derivative is not a number in general, it's a function of "direction," or more formally of a tangent vector, and smooth structures, among other things, allow you to define tangent spaces in a clean way.

2. Smooth structures allow you to define derivatives without choosing a metric. In many examples there will be a natural smooth structure but not a natural choice of metric, and you don't want to have to pick a metric if you don't need to, e.g. so that you can use more symmetries.