What dx Actually Means
You remember talking about Dx (delta x) in a precalculus course. It represents a distance along the x-axis; or, to put it another way, the difference between any two values of x. Well, dx means exactly the same thing, with one key difference: it is a differential distance, which is a fancy way of saying very, very, very small. In technical terms, dx is what happens to Dx in the limit when Dx approaches zero.
Now, when you have a quantity whose value is virtually zero, there's not much you can do with it. 2+dx is pretty much, well, 2. Or to take another example, 2/dx blows up to infinity. Not much fun there, right?
But there are two circumstances under which terms involving dx can yield a finite number. One is when you divide two differentials; for instance, 2dx/dx=2, and dy/dx can be just about anything. Since the top and the bottom are both close to zero, the quotient can be some reasonable number. The other case is when you add up an almost infinite number of differentials: which is kind of like an almost infinite number of atoms, each of which has an almost zero size, adding up to a basketball. In both of these cases, differentials can wind up giving you a number greater than zero and less than infinity: an actually interesting number. As you may have guessed, those two cases describe the derivative and the integral, respectively. So let's talk a bit more about those, one at a time.
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