I am reviewing some general relativity, starting specifically with flat space (i.e. special rel) and came across a derivation of the velocity transformation between reference frames that I had an issue with. For simplicity let us use two spacetime dimensions for now, with coordinate systems $x^{\mu} = (t, x)$ and $x^{\mu'} = (t', x')$ where $x^{\mu'}$ is boosted by velocity $v$ with respect to $x^{\mu}.$ Now, given some object moving with velocity $u = \frac{dx}{dt}$ with respect to the unprimed coord. system the velocity as viewed from the primed coordinates is $$u' = \frac{dx'}{dt'} = \frac{u - v}{1 - uv}.$$
This can be derived rigorously using the relationship between coordinates $x' = \gamma_v(x - vt)$ and $t' = \gamma_v(t -vx)$ along with the chain rule. Alternatively one could derive this from tensorial equation for four-velocity $$V^{\mu'} = \Lambda^{\mu'}_{\;\mu} V^{\mu}$$ where $V^{\mu}$ is the four-velocity in the unprimed ref. frame and $\Lambda^{\mu'}_{\;\mu}$ is the Lorentz transformation corresponding to our boost to primed coordinates.
However, I have seen a few derivations that will take the two differential 1-forms $dx' = \gamma_v(dx - vdt)$ and $dt' = \gamma_v(dt -vdx)$ then simply divide them to get $\frac{dx'}{dt'}.$ Clearly this works out, but from what I recall of differential forms you cannot in general divide two 1-forms to get a derivative, specifically when the 1-forms are acting on (paths in) an $n > 1$ dimensional manifold as is the case here. Is there any mathematical rigor to why we can do so in this case despite the fact that we can't do this in general? If so what is it? Does it have to do with the fact that both 1-forms are exact?