[bikenaga]

dx can be a differential form, but in (elementary) calculus books it's exposited this way: Suppose you have a function y = f(x) taking real numbers to real numbers, so x is an independent variable, y is the dependent variable. You define an independent variable Δx and define dx = Δx. Also define dependent variables dy and Δy by

    Δy = f(x + Δx) - f(x)  and  dy = f'(x) dx.

Then f'(x) = dy/dx. This may look like a stupid hack to make the last formula work, but actually it's a little more. If you use nonstandard analysis, you define the derivative of a function f from reals to reals by

  f'(a) = st( (f(a + Δx) - f(a)) / Δx )

where st takes the standard part of a hyperreal number and Δx is a nonzero infinitesimal. This is like the usual limit definition, without limits. Then you can use the formulas above and "dy" and "dx" are numbers, albeit hyperreal numbers.

(The "dx as a differential form" vs. "dx as a number" is probably coming from the fact that the tangent space to the reals at a real number is isomorphic to the reals, so the dual space [where dx lives] is too.)

(Calculus via infinitesimals is pretty cool; a good resource for this is H. Jerome Keisler's "Elementary Calculus" and "Foundations of Infinitesimal Calculus", both available for free: https://people.math.wisc.edu/~hkeisler/)

I second the recommendation for Barrett O'Neill's book - I used it in my differential geometry class at MIT.