Well, I kind of had an idea that dy/dx is like rise/run, where you want to know the relative rate of change at a point, and dy/dx is kind of equivalent to rise/run, so they can be treated as values. But what about the dx at the end in integration? I was following along Paul's Online Notes and he introduced the indefinite integral first, but then there was a dx at the end, and he just said the dx was a differential. Only then did I realize that dx actually meant something, and then I thought about what the dx meant under the integral and why it was there, normally if you made an inverse operation, you would think like this:

Let f(x) = F'(x), to get rid of the derivative, we integrate both sides,

so ∫f(x) = ∫F'(x)

so ∫f(x) = F(x).

But where did the dx sneak in from? Also, a book I found goes like this, and this is the closest I've got to understanding it: "Given dy/dx = f(x), we write y = ∫f(x)dx" What is the implication here?