Mathematicians will in one breath tell you they aren't fractions, then in the next tell you dz/dx = dz/dy * dy/dx
this post was submitted on 01 Jul 2025
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Have you seen a mathematician claim that? Because there's entire algebra they created just so it becomes a fraction.
(d/dx)(x) = 1 = dx/dx
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
2+2 = 5
…for sufficiently large values of 2
i was in a math class once where a physics major treated a particular variable as one because at csmic scale the value of the variable basically doesn't matter. the math professor both was and wasn't amused
Engineer. 2+2=5+/-1
Statistician: 1+1=sqrt(2)
I mean as an engineer, this should actually be 2+2=4 +/-1.
Computer science: 2+2=4 (for integers at least; try this with floating point numbers at your own peril, you absolute fool)
0.1 + 0.2 = 0.30000000000000004
comparing floats for exact equality should be illegal, IMO
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pi*pi = g
units don't match, though
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I always chafed at that.
"Here are these rigid rules you must use and follow."
"How did we get these rules?"
"By ignoring others."
is this how Brian Greene was born?
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.
Except you can kinda treat it as a fraction when dealing with differential equations
Oh god this comment just gave me ptsd
Only for separable equations
And discrete math.
clearly, d/dx simplifies to 1/x
Look it is so simple, it just acts on an uncountably infinite dimensional vector space of differentiable functions.
fun fact: the vector space of differentiable functions (at least on compact domains) is actually of countable dimension.
still infinite though
Doesn't BCT imply that infinite dimensional Banach spaces cannot have a countable basis
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If not fraction, why fraction shaped?
It was a fraction in Leibniz’s 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.
Why does using it as a fraction work just fine then? Checkmate, Maths!
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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 ∫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).
The other thing is that it's legit not a fraction.
Having studied physics myself I'm sure physicists know what a derivative looks like.
Little dicky? Dick Feynman?
1/2 <-- not a number. Two numbers and an operator. But also a number.
I still don't know how I made it through those math curses at uni.
Calling them 'curses' is apt
Is that Phill Swift from flex tape ?
Division is an operator
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.
De dix, boss! De dix!
What is Phil Swift going to do with that chicken?
The will repair it with flex seal of course
To demonstrate the power of flex seal, I SAWED THIS CHICKEN IN HALF!
We teach kids the derive operator being ' or ·. Then we switch to that writing which makes sense when you can use it properly enough it behaves like a fraction
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.
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