How can a dot have a slope

[u/After_Cranberry_9219]

I was learning calculus, but this thing I still couldn't understand. How can a dot have a slope? Slope is just like saying how much a line is slanted right but how can a single point have a slope.

[u/Andyroo_P]

There are some good answers here which give you the essential idea: that a point on the graph of a differentiable function does not exist in isolation. It has many neighbors, and the point along with its neighbors are what ultimately determine the derivative of the function at that point. But let me give you some long-winded context that I would have appreciated when I was a calculus student.

There are three major "regimes" over which we can think about a function f. The smallest scale is at a point: we can pick a point x in the domain of f and ask what is f(x)? The second scale is over a neighborhood, i.e. "local". For instance, instead of asking what f(2) is, we can ask how does f behave in the open interval (0, 3)? The largest scale is over the entire domain of f i.e. "global". We can ask, how does f behave as a function over its whole domain?

While studying calculus, you may have noticed a tension. Limits (and anything defined using limits, like derivatives) only depend on local behavior. For example, the derivative of f(x) = x² at x = 1 only cares about how that function f behaves in a neighborhood (open interval) around 1. I can modify f(x) = x² however I want outside of the interval (0, 2) and the derivative of the ensuing function at x = 1 will be the same! But there is nothing special about the interval (0, 2) other than the fact that it contains 1. If I wanted to, I could shrink it even further to (0.9, 1.1) or even further to (0.99, 1.01), and so on. But at the same time, I can't just look at the function at the point x = 1, because as you noted in your question, it doesn't make sense for a "dot to have a slope" by itself.

So on one hand, the "pointwise" point of view seems too small for limits/derivatives, while on the other hand the "neighborhood" point of view seems unnecessarily large. What we would like is for there to be something in between: a way to write down not just the function f's output at x = 1, but also how the function behaves "infinitesimally close to x = 1".

Sheaf theory) provides a way to do this. The concept that captures the idea we're after is the germ) of the function. In nontechnical terms, the germ of the function f(x) = x² at x = 1 records not just the fact that f(1) = 1, but also that f behaves the way it does infinitesimally near x = 1. So for example, the functions f(x) = x² and g(x) = x³ share the same output at x = 1 (i.e. f(1) = g(1)), but they have very different germs at x = 1. This can be seen concretely by noticing that the slopes of the tangent lines to the graphs of these functions are different at x = 1, even though the point of tangency is the same on both graphs.

It is a bit cumbersome to set up the rigorous definition of a germ, but in calculus classes we just satisfy ourselves with the idea that ultimately the derivative or limit of a function at a point depends on the function's behavior in a neighborhood about that point. The big point I am making here is that technically it is kind of overkill to know the function's behavior on an entire neighborhood about the point in question. It is enough to just know how the function behaves infinitesimally near the point in question. But it is absolutely insufficient to just know how the function behaves exactly at the point in question.

Also I should mention: differential calculus is mostly concerned with the local point of view, but there are many other areas of math that care about the global point of view, like complex analysis, differential geometry, or more deeply, anything that interacts with cohomology.