Why would you even tell in the first places derivatives are simply fractions ? They’re not, unless in some very specific physical approximations and in that case don’t try to do anything funky, sticks with the basics stuff
My understanding is they actually are fractions of things called differential one-forms[1], but even most people who can do calculus don't get to differential geometry, so the sense in which they are fractions is not commonly understood. Michael Penn explains it here https://youtu.be/oaAnkzOaNwM?si=nwNNg4pl7WW4KvIO
A 1-form is a section[1] of the cotangent bundle[2] of a manifold. In other words, a rank 1 covariant tensor field.
At any given point p on an n-dimensional manifold, a 1-form defines an n-dimensional cotangent vector (in the language of bundles[3], a point in the fiber over p).
So how do we define fractions of sections or vectors?
In the article, Baez defines fractions of 2-forms on the plane as the pointwise ratio of coefficients of a basis vector, which he can do because, as he points out, the space of 2-forms at a point on a 2-dimensional manifold is a 1-dimensional vector space (more generally, for k-forms on an n-dimensional manifold, this dimension is n choose k, so only 1 for 0-forms [functions] and n-forms).
If you are careful to represent them on the right set of variables, and apply them on the right points (what the example on the article obviously doesn't do), they pretty much behave exactly like fractions.
There are many areas of mathematics that spun from this.
Being careful is very costly, if you’re not careful you’re just approximating or being wrong. At this point we have to weigh if the tradeoff of writing it as fractions was really worth it
they refer in the beginning to physics classes, and I had the same exact experience in university. diffeq was not a prereq and yet instead of explaining the derivation of these equations, our physics professor just handwaved and said "they're basically just fractions, don't think about it too much"
[1] https://mathworld.wolfram.com/Differentialk-Form.html