A Question On Derivatives In Calculus

Question

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:

1. 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?

2. 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?

3. 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.

Answer 1

Not an expert on calculus, but I know enough to hopefully be able to help with your questions.

1. You're not necessarily looking for the slope of a point, you're looking for the slope of a curve at that particular point.

For instance, in a straight line of the format $f(x)=mx+b$, like you encountered in algebra, the slope is always the same. However, in an equation that curves, like $f(x)=x^2$, the slope or steepness of the line is constantly changing. Calculus allows you to find how steep that curve is at an exact point.

2. The beauty of calculus is that, because you're dealing with infinitely "zooming in" on a line, it allows you to actually arrive at an exact answer, not just an approximation. If you were just looking at a point very close to that original point, you'd have an approximation, but because it's infinitely close, you can treat it as the same point for the purposes of calculus.

3. Again, this allows you to find an exact slope of a line at any point on that line.

Don't feel bad -- it is a lot to take in at first and seems contradictory, but give it time, and you'll come to see the beauty and genius of the calculus!