calculus - Derivative of a single point (not a point in a function, but a point).

I know that the derivative of a line is its slope, and that a constant is a line with slope zero, so the derivative is zero.

But how does it work if instead of a constant, or a line, you had a single point? Is it just simply undefined? I tried to google for something about this but it always just returns info on "how to find the derivative at a point in a function". Which is far more useful but doesn't answer my curious question.

Let's assume you just have an X,Y graph and are given the point (5,10). The definition of a derivative includes a function so I'm guessing it's just not defined but wanted some more expert input.

Answer

You can't find an answer to your curious question because the question makes no sense. Derivatives were invented to study change. When your position changes over time the derivative is your velocity. When you climb a mountain and your height changes as you move horizontally the derivative is the slope. That's essentially the same slope you see in a graph when the value of $y$ changes as $x$ changes. There is no way to think about how an isolated point changes. On a particular mountain trail at a particular point there's a slope, but a point floating in the air has no slope.