So after three semesters of being told that dy/dx IS NOT A FRACTION and that we're not allowed to separate the dy and...

dy/dx is a limit... it's a single entity. It's misleading notation.

cos dy/dx is a limit technically if im not mistaken so separating that "fraction" actually makes no sense

Yeah sorry I was thinking of ∆x

Leibniz was a mistake

There exists equally rigorous constructions of the derivative where dy represents infinitesimal change and /dx represents division by an infinitesimal.

not if you want to extend to multivariable analysis, particularly if you try and construct a reasonable change of variables that reproduces the jacobian result. Unless of course you just change gears and abuse the language of differential forms out of nowhere.

hey you sound like u know what ur talkin about

help a brainlet out
differential dy/dx is just a limit hunh
how do u do a separable differential equation if u treat differentials like that

I would guess it's just a notation of sorts.
dx just means with respect to x

>how do u do a separable differential equation if u treat differentials like that
Did you not read ?

Ultimately, multiplying by dx is just a heuristic... a "shortcut" for doing it the "real way." It has no actual valid mathematical meaning (that is, if we treat dy/dx as a limit).

yes but how would i do it the real way? i don't see how the stackexchange answer is suitable as he seems to play with and separate differentials as well

isn't the first derivative of (xy)^2 equal to 2x^2y + 2xy^2 ?

They didn't differentiate, they just rewrote it in an equivalent form. (xy)^2 = (x^2)(y^2)

ye i'm just a mere brainlet thanks for pointing it out.

Once you proof rigorously that your symbol manipulation is valid, you can do it.
Same with integration by substitution and differentiation of compositions.
It's easier for people to remember that way

It's just notation, but you should be comfortable proving it yourself.

...

>implying you can't separate one limit into two
[math]\lim_{x -> x'}{\dfrac{y(x)-y(x')}{x-x'}}=\dfrac{\lim_{x -> x'}{y(x)-y(x')}}{\lim_{x -> x'}{x-x'}}[/math]

that's wrong

it's a fraction!
[eqn]\frac{dy}{dx} = (xy)^2 [/eqn]
now cancel the "d"s
[eqn]\frac{y}{x} = (xy)^2 [/eqn]
or
[eqn]y = x^{-3} [/eqn]

tbf i have yet to see a case where this wouldn't work

the fuck is wrong with you faggots, looks at OP's picture. [math]x^2 dx \neq x [/math] Simple typo but you guys should have noticed.
Also as an engineer, I have yet to encounter a differential equation I can not solve by selecting a finite difference approximation and rewriting as a recurrence relation. The ability to go back and forth between dx notation and some finite difference is enough to make mincemeat of anything encountered in a college diff equ class.

U can manipulate integratipn by parts to prove 1=0.

dy/dx is not a fraction. BUT dy=(dy/dx)*dx

That still not a fraction. It's defined as the best linear aproximation, you still need to prove you can manipulate it.

Can you read? I said dy/dx is not a fraction. It's a slope. dy=(dy/dx)*dx. Infinitesimal change in y is equal to the slope times infinitesimal change in x.

>infinitesimal
Define this concept properly ypu retard

>infinitesimal
small change. Is it really that hard to grasp?

Define small

the distance between two points on a curve where the distance can be reasonably approximated as a straight line.
>inb4 define reasonably approximated
It depends on the application

One must learn first the concept of derivative, and understand it well.

Then one fucks around with dx and dy because it's literally useful, specially in Physics if one thinks of what this means physically (an infinitesimal rate of change of some quantity).

You can do the variable separation method with additional intermediate steps. For example, you take the first line, you integrate both sides of the equation in x. then you get the 2nd line.

This is also useful once you get to partial differential equations, since it will actually lead you places.

But when it's one variable, come on.

>y'(x') = 0/0
genius

No you retard, we are talking about proper mathenatical definition. I'll give you a definition brainlet. The diferential is the best linear aproximation of a function at a point. That means the error function over the distance between two points tends to 0 as you aproximate the point. By that means a limit which is well defined.

He's an engineer, ignore him.

>the best linear aproximation of a function at a point.
>the distance between two points on a curve where the distance can be reasonably approximated as a straight line.
Yep these two statements are completely different. And since you asked me to properly define infinitesimal, how about you properly define the error function for me?

See Taylor's theorem faggot.

You could have looked up infinitesimal, but you instead asked me to define it. I'm just asking the same of you. No need to get butthurt.

Its also not propely deined as "small" you cuckgineer faggot.

are you saying infinitesimals aren't small?

Im saying that small is not a rigorous mathematical concept. That's why there are a ton of equivalent definitipns that range from algebra to analysis

what makes a concept mathematically rigorous?

>You can sometimes pretend, this is one of the few examples.
just say you're doing a subtle change of basis already

It is an infinitesimal fraction, some of the time. Thinking of it as a fraction is the reason seperation of variables works with linear ODEs and why arc length works in n => 2

That's actually a rather complicated question. It's so hard to pinpoint that ypu have to bring historical concepts to some proofs. But generally, something is mathematically rigorous id you could derive it from first principles using a particual set of symbols called a "language". In most cases this principles are ZFC set theory using first order logic I think. That's why its important to write small with the delta-epsilon definition and why inequalities are crucial.

holy shit faggot just fuck off and kill yourself

Thanks. I still haven't gotten a rigorous definition for the error function other than a vague reference to Taylor's theorem.
Stay mad

You should infer it from the name... It's a function that by definition gives you how much your linear aproximation deviates from the value of ypur function. And no, it's not given explicitly, thate important part is that it tends to 0 "rapidly" as you get close to the point. Literally just read the wikipedia entry on Taylor's theorem brainlet

>You should infer it from the name
And yet you can't infer what small means by the name? This is my point.
>it's not given explicitly
Then how can it be apart of a rigorous definition?
>important part is that it tends to 0 "rapidly" as you get close to the point.
What do you mean by "rapidly"? It depends on the application, as I said in You're just mad that I turned out to be right from the very beginning. Try thinking critically before you criticize others.

It's just combining intuition (when it's sensible, you can think of the derivative as approximately the change in a function divided by its change in variable) and the fact that it works as claimed due to the fact that
[math]
\int g(f(x)) \frac{df}{dx} dx = \int g(f) df \,\,\,.
[/math]
Extend to a class of first order separable differential equations as:
[math]
y'(x) = f(y(x))*g(x)
\implies
\frac{y'(x)}{f(y(x))} = g(x)
\implies
\int \frac{y'(x)}{f(y(x))} dx = \int g(x) dx \,\,\,.
[/math]
Then just apply the usual result and you get that the 'dy/dx' separate into an integration over y (symbolic 'dy') and an integration over x (symbolic 'dx').

>And yet you can't infer what small means by the name?
You fucking brainlet, small is a relative concept. For every "small" distance, you could multiply it by 0.00000000000000000000000000000000000001 to make it even smaller. And you can repeat this recursively. So at what point do you stop and say that a distance is "small" enough? Whatever you pick will be completely arbitrary, anyway, since there are an infinite number of real numbers between any two real numbers. Saying "it's just a small change dude lol" is not mathematically useful.

>small is a relative concept.
>important part is that it tends to 0 "rapidly" as you get close to the point.
"Rapidly" is a relative concept too. And now that I think about it, so is "close". At least my definition only relies on one relative concept.

Nigger, I'm not going to write all the mathenatical notation. Literally just check the entry for Taylor's theorem ffs.

How

Alright, this is my first time on this board, but are you people seriously talking about advanced class in highschool/early-college level calculus this seriously?

I think this may also be my last time on this board, Jesus Christ, bunch of autists and 18 year olds.

It's not a fraction, all you are doing is integrating with respect to X and using the chain rule.

Went to wikipedia and ctrl+f "error function" and got nothing. Also noticed that "The exact content of "Taylor's theorem" is not universally agreed upon." Why would you demand a rigorous definition from me when you are too lazy to be rigorous yourself?

>You can sometimes pretend

And Veeky Forumstards continue to pretend science is as rigid as they claim it to be.

F-f-fuck off, Veeky Forums!

We did that in leaving cert applied maths, bro. Yeah, it makes no sense for the first few years, but you work out the logic of it.

[math]dy[/math] and [math]dy[/math] are differentials. What they are is a representation of the arbitrary slope of x at that point.

However, this by itself is not defined. [math]x[/math] is just a real number, so [math]dx[/math] has no meaning in and of itself. However, [math]\frac{dy}{dx}[/math] does have meaning, because we have a relationship between [math]x[/math] and [math]y[/math] through the function [math]f(x):=y[/math]. [math]\frac{dy}{dx}[/math] tells you how fast [math]x[/math] is changing with regard to [math]y[/math] at any point [math]x[/math] where [math]\frac{dy}{dx}[/math] is defined. If you understand

At [math]x[/math], [math]\frac{dy}{dx}[/math] is defined as [math]\lim_{t\to x}\frac{f(t)-f(x)}{t-x}[/math] when this limit is defined. Which, if you think, means we're taking smaller and smaller chunks of [math]f(x)=y[/math] compared to smaller chunks of [math]x[/math]. So if you want to think in terms of infinitesimals, (which only should be used as a heuristic), we have [math]\frac{dy}{dx}\approx\frac{\Delta y}{\Delta x}[/math].

In context, it definitely can be inappropriate to move a differential. In calculus, with a rigidly defined [math]f(x):=y[/math], [math]\frac{dy}{dx}[/math] has a definite meaning, particularly it is a function [math]f'(x)=\frac{dy}{dx}[/math] and takes explicit number values at every [math]x[/math]. To "split up" [math]\frac{dy}{dx}[/math] would take a number and try to replace it with differentials. Algebra simply doesn't work that way, it's a nonsense gesture.

However, in the case of a differential equation, we're attempting to find a function which works as a solution to an equation. As such [math]\frac{dy}{dx}[/math] has not taken on an explicit meaning. Therefore it's completely appropriate to manipulate them algebraically.

If you wanted to understand what was told to you, you would have realized pretty quickly that the "reminder" is what he referred as an error function.

autism

>after three semesters

>americuck education

I'm an engineer, faggot, what did you expect?

It's not a fraction in the sense you can't just go and multiply it with other shit you've got around and solve for d or whatever. It is a fraction in the sense that it's an infinitely small change in y over an infinitely small change in x. Just use fucking common sense when doing math, jesus.

>3 semesters
>dy/dx ain't a fraction
Smh

Faulty. Life is messy.

>can't understand such a simple concept
>can't google for an explanation
>fucking christ engineers are retarded

...

>IS NOT A FRACTION
It isnt, it is just notation that alows you to do this things.

If you understood what dy/dx meant you would understand why you can do this.

I've never understood differential equations beyond 1st order ODEs

It's beginning to catch up with me now since I have a module at university which is mainly based on differential equations. Hopefully by the end of this year I will understand them

You can't divide by differential forms.

Listen, OP. Some of the explanations so far are good -- others are cancer.

No, it's not really a fraction, mathematically speaking. Intuitively speaking, dx or dy, represents some infinitesimally small difference in y or x, and can be thought of as such.

If you want a rigorous, mathematical explanation for why you can "multiply" by dx, recall the chain rule.

>he doesn't know about infinitesimals

how cute, it's a brainlet

differentials are not infinitesimals.

"an indefinitely small quantity; a value approaching zero."

What are you on about. I'm using the term in the classical sense

there is no such thing as an indefinitely small quantity. differentials are defined in terms of limits. they are different ideas.

Please define the size of a point in space. I'll give you a hint: it is infinitesimally small.

a point has zero measure.

But it exists in space, does it not?

the length of a point is zero.

Measure is a well defined concept

Who told you it wasn't a fraction?

So please explain what is wrong with my use of the word "infinitesimal"

there is no such thing as an infinitely small numbers. [math]\frac{dy}{dx}\approx\frac{\Delta y}{\Delta x}[/math] is true but it's never equal for any [math]\Delta[/math]. You must take the limit, which is expressly different than the infinitesimal.

It may seem like a "trivial" detail to you, but infinitesimals are wrong enough to cause serious problems in mathematics, they break and make things contradictory. I know that might not matter to someone like you, but an engineer's perspective on math does not matter.

How do you faggots not know this and end up treating derivatives as fractions?

'Litchrully' fucking kill yourselves you homosexual penis-loving engineer cucks.

underrated

limits go through divisions retards.
Lim a/b = Lim a / Lim b

>Lim a/b = Lim a / Lim b
>Lim x-->0 (2x)/(2x) = Lim x--> 0 2x / Lim x --> 0 2x
>Lim x-->0 (2x)/(2x) = 0/0
Huh.

AFTER 3 SEMESTERS OF BEING TOLD AT PRIMARY SCHOOL THAT I CANT TAKE AWAY LARGER NUMBERS FROM SMALLER NUMBERS IM NOW GETTING TOLD THAT IM ABLE TO DO THIS?

FUCKING CHRIST TEACHERS R RETURDED