I am just starting Calculus, and am very confused about the idea of derivatives. I get why we need derivatives, because we need to find the slope of a single point in a line, and our conventional method of finding a slope (y1-y2/x1-x2) won't work if there's only one point.
But there are many things about this that confuses me:
How can there be a slope if there's only one point? The whole premise makes no sense to me, finding the slope of a single point. To have a slope, you need a line, but how can you have a line with only one point? So how is it possible to find the slope of one point; and what would that result even mean? What does the slope of a point mean, when only lines can have slopes?
Are the values found by derivatives just approximations? Because from what I read, the solution to the problem above is it find a point that is infinitesimally close, then find the slope of that? So isn't that an approximation?
I understand the "solution to this problem when calculating is to shift delta x to 0 as seen here: . But then if we shift delta x to 0, then the base of the fraction would be 0! Then how would it be possible to calculate?
- Does that also mean, we can never find the actual slope of a point because all the results we find are just slopes of lines that are very small, but not slopes of actual points?
Can you try to explain this not too rigorously, at the level of someone just starting Calculus? Thanks.