So Why Do We Treat It That Way?

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

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".

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

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.

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

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.

Thats gotta be the most anti-algorithm title ever

Hi Aerospace Engineering Lecturer here. We are a special breed in Engineering where, we use the egghead's mathematics but are 100% utterly honest and forgivingly blunt about it. When the power hour duo of the Issacs (Barrow and Newton) both both began developing the more modern derivative by limits definition, it has been hailed as this beautiful thing in math with the complicated name that gives students headaches. In engineering we teach it way more bluntly. The Issacs used the most beautiful tool in human beings tool kit. Their inner morons 😆Like actually think about it... "I cant measure the slope of a curvey line! SOD IT WE DOIN IT LIVE BOIS!" Proceeds to slap a straight line on the curve and force it into submission My personal theory as to why, over the years, we've said that dy/dx is not a fraction is simple... Its the same reason we decided to call the theory in beam bending that allows you to just add 2 beam cases together (literally the theory, "Addition go brt") the "Principle of Superposition". We don't like admitting our "Smart" ideas are actually much dumber than we pretend. We are the same species that came up with 3 laws of thermodynamics... then found a 4th but it was the most important and went "Should we change the 1st to 2nd, 2nd to 3rd, 3rd to 4th and make this new one 1st? Sod it call it the 0th law... dy/dx is not a notation, I adamantly will fight any mathematician on this... dy/dx is the brutally honest admission of what we have done to get our fancy differential equations... We slapped a straight line onto a curve and measured its gradient after making the line infinitely small... And the equation for a gradient of a straight line? y2 - y1 / x2 - x1 aka.... ChangeY/ChangeX aka dy/dx.... We just really REALLY don't like thinking about the fact that the entirety of one the most revolutionary and culturally seen as "most difficult and academically impressive" math we've done as species boils down to "hehe line on curve but small hehehehe". In engineering we arn't afraid to admit that we, beyond all our fancy text books and our latin-based names... are still just stupid monkey brains who have been bodge jobbing our way to success for centuries 😅 dy/dx is a fraction, we should start accepting that...

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

I'm still not satisfied. The only complication seems to be because dy and dx are infinitesimal they have no useful value outside of their relation to each other. But treating dy/dx as a fraction and performing regular valid manipulations never breaks that relationship. As for people saying "well you can't cancel the d's", including my first year maths lecturer, that is completely stupid, of course not, the d's are not separate values or variables. (Which is why it is preferable to keep your d's roman or unitalicised btw.)

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). Wilipedia calls it Cauchy's approach (https://en.m.wikipedia.org/wiki/Differential_of_a_function).

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

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).

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

I think of it as an infinitesimal change divided by another infinitesimal change.

Bro really just proved the entirety of physics, my my

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

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

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