Mathematicians will in one breath tell you they aren't fractions, then in the next tell you dz/dx = dz/dy * dy/dx

The thing is that it's legit a fraction and d/dx actually explains what's going on under the hood. People interact with it as an operator because it's mostly looking up common derivatives and using the properties.

Take for example ∫f(x) dx to mean "the sum (∫) of supersmall sections of x (dx) multiplied by the value of x at that point ( f(x) ). This is why there's dx at the end of all integrals.

The same way you can say that the slope at x is tiny f(x) divided by tiny x or d*f(x) / dx or more traditionally (d/dx) * f(x).

clearly, d/dx simplifies to 1/x

I found math in physics to have this really fun duality of "these are rigorous rules that must be followed" and "if we make a set of edge case assumptions, we can fit the square peg in the round hole"

Also I will always treat the derivative operator as a fraction

If not fraction, why fraction shaped?

Derivatives started making more sense to me after I started learning their practical applications in physics class. d/dx was too abstract when learning it in precalc, but once physics introduced d/dt (change with respect to time t), it made derivative formulas feel more intuitive, like "velocity is the change in position with respect to time, which the derivative of position" and "acceleration is the change in velocity with respect to time, which is the derivative of velocity"

Except you can kinda treat it as a fraction when dealing with differential equations

1/2 <-- not a number. Two numbers and an operator. But also a number.

It was a fraction in Leibniz’s original notation.

The world has finite precision. dx isn't a limit towards zero, it is a limit towards the smallest numerical non-zero. For physics, that's Planck, for engineers it's the least significant bit/figure. All of calculus can be generalized to arbitrary precision, and it's called discrete math. So not even mathematicians agree on this topic.

Why does using it as a fraction work just fine then? Checkmate, Maths!

This very nice Romanian lady that taught me complex plane calculus made sure to emphasize that e^j*theta was just a notation.

Then proceeded to just use it as if it was actually eulers number to the j arg. And I still don’t understand why and under what cases I can’t just assume it’s the actual thing.

It's not even a fraction, you can just cancel out the two "d"s

Look it is so simple, it just acts on an uncountably infinite dimensional vector space of differentiable functions.

Having studied physics myself I'm sure physicists know what a derivative looks like.

But df/dx is a fraction, is a ratio between differential of f and standard differential of x. They both live in the tangent space TR, which is isomorphic to R.

What's not fraction is \partial f / \partial x, but likely you already know that. This is akin to how you cannot divide two vectors.

When a mathematician want to scare an physicist he only need to speak about ∞

I still don't know how I made it through those math curses at uni.