Listen here, Little Dicky

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

Brah, chain rule & function composition.
Have you seen a mathematician claim that? Because there's entire algebra they created just so it becomes a fraction.
Also multiplying by dx in diffeqs
- vietnam flashbacks meme
(d/dx)(x) = 1 = dx/dx
This is until you do multivariate functions. Then you get for f(x(t), y(t)) this: df/dt = df/dx * dx/dt + df/dy * dy/dt
Not very good mathematicians if they tell you they aren't fractions.

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
- Found the engineer
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.
yea, essentially, to me, calculus is like the study of slope and a slope of everything slope, with displacement, velocity, acceleration.

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

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

"d"s nuts lmao

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

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.

If not fraction, why fraction shaped?

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

When a physicist want to impress a mathematician he explains how he tames infinities with renormalization.
Only the sith deal in ∞
- ....and Buzz Lightyear

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.

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.

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}
e^𝘪θ is not just notation. You can graph the entire function e^x+𝘪θ 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^𝘪θ). The complete version is:
e^x+𝘪θ := e^x(cos(θ) + 𝘪sin(θ))
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.
- I'm aware of that identity. There's a good chance I misunderstood what she said about it being just a notation.
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.

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.
- it's legit a fraction, just the numerator and denominator aren't numbers.

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

Chicken thinking: "Someone please explain this guy how we solve the Schroëdinger equation"

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 ?

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)

Little dicky? Dick Feynman?

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.

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!

De dix, boss! De dix!

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

Software engineer: 🫦

Headache for mathematicians

https://youtube.com/shorts/WSFkDNXOpMk

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.