So Why Do We Treat It That Way?

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[vivada]

"Nooo don't treat dy/dx as a fraction it only works 100% of the time"

[naytte]

The reason it works is because it IS intended to be a fraction. Indeed, Leibniz thought of dy/dx as a ratio of infinitesimals. So it's by no means some coincidence that this works, like some people seem to think. Actually, theories of infinitesimals that support calculus do exist, as shown by Abraham Robinson in the 1960s with his hyperreal numbers.

If you are interested, there is a good entry-level book by H. Jerome Keisler called "Elementary Calculus: an infinitesimal approach".

[ManilockPi]

If not a fraction, why fraction shaped?

[eliteteamkiller]

Whenever the d is STRAIGHT and FIRM, you can cancel. Whenever the d is flaccid — I mean curved, you know, partial derivatives — then you can’t cancel.

[alessandrocoopman]

In physics it's common practice to treat Leibniz's as a fraction.

[ingGS]

I am an Engineer, I see dy/dx and chances are treating it as a fraction is part of my solution.

[mikey-hmdt]

Thats gotta be the most anti-algorithm title ever

[jmcsquared]

So then, why does the chain rule work? Turns out, the proof of the chain rule is verbatim (with a well-definedness check so that we don't divide by zero) going to the difference quotient and performing a cancelation of fractons before taking a limit.

Yes, dy/dx is a fraction. It's just an infinitesimal fraction; that requires developing careful intuitions for it. Ones which btw do not hold in multiple variable contexts: partials behave quite differently sometimes from their one-dimensional cousins.

[professorpoke]

"dy/dx is a fraction and most probably will always be a fraction to me. You can't change my mind."

[maringenov]

We treat it as a fraction b/c it IS in fact a fraction: it's the ratio of two functionals, called differentials - these are sections of the cotangent bundle of the reals (the fibers just happen to be canonically the real line itself). If you think about it a little bit, the way to capture the original intuition of infinitesimal quantities in a rigorous way is precisely by using vectors (and consequently Linear Algebra) b/c on the one hand vectors are points, but on the other they are also quantities with magnitude and direction. Intuitively, if you let the magnitude get infinitesimally small, you are still left with the direction, so an abstract direction corresponds to an abstract (directed) infinitesimality (directed b/c for example it can have a sign).

[live_free_or_perish]

According to Leibniz, the quotient of an infinitesimal increment of y by an infinitesimal increment of x. So it is a fraction

[matthewmason]

are there any examples showing that treating dy/dx as a fraction doesn’t always work?
edit to clarify: with ordinary derivatives, not partial

[slava]

Looked in my calculus lectures and also in Wikipedia:
The are different approaches to define differential operator.
Our university stated for us that there are derivatives (y'(x)).
Then that there's differential (d(y) = dy = y'(x)*dx).
And there arises identity: dy/dx = y'(x).

[OBGynKenobi]

But it's essentially the slope, which is a fraction.

[smesui]

dy/dx has been lost in the shuffle. dy/dx is limit ( lim ) A->0 of Ay/Ax where Ay = ( y - y1 ) ; Ax = ( x - x1 ). That's it !

Further elaboration:

dy/dx may be considered irrational. An irrational quantity is a real quantity which can't be represented as a ratio of two integers ' n ' the numerator and ' d ' the denominator as n/m, m =/= 0 .

[sieni]

This notation gets intoition from the world of differential forms where 0-forms are functions and taking exterior derivative of a 0-form f yields 1-form df=f'dx, where dx denotes the exterior derivative of the x-coordinate function (by coordinate function I mean we can generalize to R^n). So the notation comes from the equality presented above (in R^n sum of all partial derivatives df=\sum_i^n \partial_i fdx_i and is obviously generalized to manifolds).

[evilotis]

oh, so you include actual Brilliant content in your subject matter, meaning i can't just skip the sponsored part of the video? that's .... that's brilliant, damn you

[Thesaddestmomentinourlives]

isn't it just a ratio? I mean we have differentiable function's rise formula:
Δy = AΔx + o(Δx) where differential is defined by AΔx and it's just a product of A and Δx
after doing some tricks with dividing by Δx and taking a limit we're getting
f'(x_0) = A, so f'(x_0)Δx = dy
and by this formula we can see that Δx = dx(not Δx ≡ dx) and thus f'(x) = dy/dx? I might be missing something, at least I think so. What did I get wrong?

[TomKorner-yelf]

Bro really just proved the entirety of physics, my my

[v.r.kildaire]

I mean from first principles it really is just an infintesimal fraction, This is where all foundational derivative rules... including chain rule, come from.