Software engineer: π«¦
this post was submitted on 01 Jul 2025
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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 β«f(x) dx to mean "the sum (β«) 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).
The other thing is that it's legit not a fraction.
it's legit a fraction, just the numerator and denominator aren't numbers.
No π
try this on -- Yes π
It's a fraction of two infinitesimals. Infinitesimals aren't numbers, however, they have their own algebra and can be manipulated algebraically. It so happens that a fraction of two infinitesimals behaves as a derivative.
Ok, but no. Infinitesimal-based foundations for calculus aren't standard and if you try to make this work with differential forms you'll get a convoluted mess that is far less elegant than the actual definitions. It's just not founded on actual math.
1/2 <-- not a number. Two numbers and an operator. But also a number.
In Comp-Sci, operators mean stuff like >>, *, /, + and so on. But in math, an operator is a (possibly symbollic) function, such as a derivative or matrix.
Youre not wrong, distinctively, but even in mathematics "/" is considered an operator.
https://en.m.wikipedia.org/wiki/Operation_(mathematics)
oh huh, neat. Always though of those as "operations."
We teach kids the derive operator being ' or Β·. Then we switch to that writing which makes sense when you can use it properly enough it behaves like a fraction
If not fraction, why fraction shaped?
clearly, d/dx simplifies to 1/x
Having studied physics myself I'm sure physicists know what a derivative looks like.
Why does using it as a fraction work just fine then? Checkmate, Maths!
It doesn't. Only sometimes it does, because it can be seen as an operator involving a limit of a fraction and sometimes you can commute the limit when the expression is sufficiently regular
Added clarifying sentence I speak from a physicists point of view.
I still don't know how I made it through those math curses at uni.
Calling them 'curses' is apt
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 donβt understand why and under what cases I canβt just assume itβs the actual thing.
e^πͺΞΈ^ is not just notation. You can graph the entire function e^x+πͺΞΈ^ across the whole complex domain and find that it matches up smoothly with both the version restricted to the real axis (e^x^) and the imaginary axis (e^πͺΞΈ^). The complete version is:
e^x+πͺΞΈ^ := e^x^(cos(ΞΈ) + πͺsin(ΞΈ))
Various proofs of this can be found on wikipeda. Since these proofs just use basic calculus, this means we didn't need to invent any new notation along the way.
It legitimately IS exponentiation. Romanian lady was wrong.
It is just a definition, but it's the only definition of the complex exponential function which is well behaved and is equal to the real variable function on the real line.
Also, every identity about analytical functions on the real line also holds for the respective complex function (excluding things that require ordering). They should have probably explained it.
She did. She spent a whole class on about the fundamental theorem of algebra I believe? I was distracted though.
Let's face it: Calculus notation is a mess. We have three different ways to notate a derivative, and they all suck.
Calculus was the only class I failed in college. It was one of those massive 200 student classes. The teacher had a thick accent and hand writing that was difficult to read. Also, I remember her using phrases like "iff" that at the time I thought was her misspelling something only to later realize it was short hand for "if and only if", so I can't imagine how many other things just blew over my head.
I retook it in a much smaller class and had a much better time.
I've seen e^{d/dx}
Mathematicians will in one breath tell you they aren't fractions, then in the next tell you dz/dx = dz/dy * dy/dx
Not very good mathematicians if they tell you they aren't fractions.
Have you seen a mathematician claim that? Because there's entire algebra they created just so it becomes a fraction.
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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
I always chafed at that.
"Here are these rigid rules you must use and follow."
"How did we get these rules?"
"By ignoring others."
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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"
Possibly you just had to hear it more than once.
I learned it the other way around since my physics teacher was speedrunning the math sections to get to the fun physics stuff and I really got it after hearing it the second time in math class.
But yeah: it often helps to have practical examples and it doesn't get any more applicable to real life than d/dt.
I always needed practical examples, which is why it was helpful to learn physics alongside calculus my senior year in high school. Knowing where the physics equations came from was easier than just blindly memorizing the formulas.
The specific example of things clicking for me was understanding where the "1/2" came from in distance = 1/2 (acceleration)(time)^2 (the simpler case of initial velocity being 0).
And then later on, complex numbers didn't make any sense to me until phase angles in AC circuits showed me a practical application, and vector calculus didn't make sense to me until I had to actually work out practical applications of Maxwell's equations.
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It was a fraction in Leibnizβs original notation.
And it denotes an operation that gives you that fraction in operational algebra...
Instead of making it clear that d is an operator, not a value, and thus the entire thing becomes an operator, physicists keep claiming that there's no fraction involved. I guess they like confusing people.
Division is an operator
Except you can kinda treat it as a fraction when dealing with differential equations
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