>being past Calc 1
>STILL writing "dy/dx" meme notation instead of just y' or Newton
I can understand this if you're learning derivatives for the first time, since it illustrates what you're doing, but there's no reason anyone familiar with the concept should should be wasting ink or graphite by writing all of this out. You may as well write "change in y over change in x."
How's Calc 2 going for you, OP?
I thought Newton's notation is only used to represent the change in time of something.
I like more Leibniz's notation. Euler's also fine.
Is /sci/ filled with nothing but high school students and college freshmen?
>>7663504
According to historic records, Newton's notation was so shit that because England wanted to push english supremacy in science they taught newton's notation in school and that basically slowed people down. Meanwhile the countries and people using Leibniz's notation would learn and understand calculus faster and would be the first ones to come up with proofs and new results for the following decades.
I have used Lagrange's notation, Euler's notation and Leibniz's notation. Lagrange's is my favorite but professors now make us dy/dx for a reason.
A lot of simple integral calculus problems need you to see it as dy/dx and manipulate that.
Furthermore, many unintuitive proofs literally are not possible without seeing it as dy/dx. I'm sure you've heard:
>Multiply both sides by dx
I like Euler's notation best but only if you write D_x or something to indicate the variable with respect to which you are differentiating.
Leibniz is pretty based as well, but when teaching it you need to stress the fact that d/dx is an operator and that you don't think of it as a fraction (except when you do)
>>7663523
>I thought Newton's notation is only used to represent the change in time of something.
It is. It's also only physicists who use it because the only thing they measure is their perception of the size of Newton's penis. (physicists are very slow so the penis appears much larger than it actually is)
>>7663525
mostly ya
>>7663504
>he hasn't reached the point where there is more than 1 independent variable
Agreed, OP. Leibniz notation is like the Calculus version writing out tally marks and counting them to add and subtract.
>>7663545
You should be able to look at f' and immediately know it means df/dx. Then when you multiply, know what happened (I multiplyed f'(x) by infinitesimal x.) If you need every step illustrated, you're not doing nonintuituve proofs anyway.
>Not usign Liebniz for partial derivatives
Off yourself m9
>>7663504
Can you show me how to use y' notation to show partial derivatives? I want to be a top tier mathematician like you OP.
f'(x) is better than y' for beginners, but dy/dx is a good intro notation for calc so you can understand why we use similar notation or differentials and integration.
>>7663641
Z'/a', Z'/b', Z''/b'^2, Z''/a'b'
>>7663504
Euler's operator notation is the best notation
>>7663608
>You should be able to look at f' and immediately know it means df/dx
What if f is a function of x,y,z, and t?
>>7663641
[math] f_{x} = ∂f/∂x [/math]
[math] f_{xx} = ∂^2 f / ∂x^2 [/math]
[math] f_{xyz} = \frac{ ∂^3 f } { ∂x ∂y ∂z } [/math]
>>7663666
Causes problems when your functions already have subscripts.
But you knew that, Satan.
>>7663677
>only working with one variable
>>7663659
I agree.
>>7663682
No idea where you're getting that, since subscripts generally get used any time you index a collection of functions.
>>7663666
That notation is complete shit.
ᵀʰᶦˢ ᵖᵒˢᵗ ʷᵃˢ ˢᵖᵒᶰˢᵒʳᵉᵈ ᵇʸ ᴹᶦᶜʳᵒˢᵒᶠᵗ'ˢ ᵂᶦᶰᵈᵒʷˢ 10™
>>7663696
[math]∑ {}_{n}f = F [/math]
[math]∑ {}_{n}f_{x} = F_{x} [/math]
says the fuck who hasnt done real analysis
>>7663786
Who says they're being summed?
Who says there are countably many?
>>7663608
>>7663662
This pretty much.
Now that Newton's notation has pretty much been assumed to mean with respect to time, each has their own place:
Leibniz' for starting out (both scalar and vector calculus, partials, etc.) because of its similarity to the limit definition of the derivative, Newton's as shorthand for rates or Lagrange's for the more general cases.
Euler for derivatives as operators, and for when you introduce Laplace Transforms I guess.
But of course, it doesn't really make a difference at the end of the day. Write whatever the fuck you want as long as its not ambiguous.
>>7663504
When you actually get to doing maths, rather than early undergrad exercises, you'll realise it's rather convenient for eigenproblems to have an operator notation which can be placed independently of the thing on which it acts.
>>7663833
>Write whatever the fuck you want as long as its not ambiguous
Most sensible post in the thread.
>>7663804
HE just did
>>7663504
dy/dx is superior as it indicates what it actually means; being a quotient and everything.
>>7663579
>operator and that you don't think of it as a fraction (except when you do)
I prefer thinking of it as a fraction, except when you can't.
>>7663653
Ayy factor spaces
I bet he doesn't even know what Df means. How cute.
Do you know what d/dx buys you OP? Do you know what the derivative really is?
Please, enlighten us.
>>7663586
>Einsteinian penis joke
i kek'd
>>7664178
It's a quotient of 2 infinitesimals. Learn some nonstandard calculus.
>>7663804
>Who says there are countably many?
Who said n∈ℕ?
>Who says they're being summed?
Take the difference of the sequence or log the product and make it a sum
>>7664178
Mathfags are probably going to rage but here goes:
So the derivative is just rate of change which is just the slope of the tangent line which is just rise/run which is [amount of y]/[amount of x]. So dy and dx are just chunks of y and x respectively that comprise the the slope of the tangent.
So in other words, the tangent line of an curve has the equation: y = (dy/dx)x + c.
>>7664203
what is an infinitesimal in this context?
i'm trying my best. my father was an astrophysicist but he died so i can't ask him for help
>>7664178
The idea is that we get a change in y divided by a change in x.
The change of a function with respect to a variable can be approximated near a point with
[eqn]\frac{dy}{dx} \approx \frac{\Delta y}{\Delta x}[/eqn]
Where [math]\Delta y = y - y_0[/math] and [math]\Delta x = x - x_0[/math]
>>7664217
Sorry I forgot the rest of my post:
As [math]y_0[/math] approaches [math]y[/math], and likewise in [math]x[/math], we get something closer and closer to our derivative. Since [math]\Delta y \wedge \Delta x \in \mathbb{R}[/math], it makes sense to treat their limits like real numbers, in a sense. In later courses however, we learn the true meaning of derivatives and what they mean as operators.
I'm not going to give away all the fun, take some more math classes.
>>7664210
that helps a lot actually.
thank you for explaining it to me that way, i think i understand it better now
so, dy/dx is a ratio of ... the "infinitesimally small" parts of y and x? i understand the limit definition of the derivative pretty well, but it's hard to translate the limit definition into a ratio of y to x, unless you, uh, define y as delta x, i guess? right. duh.
>>7664226
no, wait, y is a function of x. not delta x. sorry, tired
>>7664229
>>7664226
seems like you got it anon. get some sleep and just study it tomorrow.
I'm a maths major working through second year of undergrad. took so much calculus last year, taking proofs right now, and i'm doing complex analysis and systems of ODEs next.
I'm trying to major in calculus, if at all possible. just tons of calculus and geometry. i'd be ok with that.
![Derivative_GIF[1].gif](https://i.imgur.com/AwdE8AD.gif "Derivative_GIF[1].gif")
>>7664233
yes, this is our main motivation for the derivative. clearly, things get a bit more involved once you want to find all such ratios such that they are the slope of the function at a point. that's where the definition of the derivative and the limiting process come in. the attached picture is a GIF from wikipedia showing exactly what we mean by this limiting process.
>>7663504
>single variable functions
cherish those days while you can my friend
>>7664213
y(x+dx) = y(x)+dy
dy = y(x+dx)-y(x)
divide out by a dx
kill any remaining infinitesimal term since they are infinitesimal compared to any real number.
>>7664226
>"infinitesimal number" hand-waving explanation
>not delta-epsilon limit definition
shiggy
>>7664265
>hand-waving
Get with the time grandpa
so whats the rigorous mathematical reason you can 'multiply by dx'?
Back in my first semester the profs told us that 'it works, dont let any mathematician see it, you will later learn why it works'.
i've never learned why. i read something about differential forms but i never really understood it.
Any anon willing to give me some insight?
>>7664301
>i
>>7664322
fuck off
>>7664331
You must be over the age of 18 to browse this site. Please leave until you are of age.
>>7664333
so you cant answer my question then?
>>7664345
dx and dy are measures
happy?
>>7664301
If you work in a non-standard analysis setting dx can be seen as a well-defined infinitesimal small number on which you can perform certain operations like multiplication. In this setting this is all fine and works well, but thats not really what people like
>>7664269
>>7664203
do and what you often learn in the not so rigouris math courses. They pretty much just treat the real functions as if they were on the hyperreals and use the tricks from that setting, which often work because for "simple" functions pretty much everything is equivelent. Thats not a very clean way to do things (and there are cases where this approach doesnt work), but its often more easy or intuitive so thats why people teach this, but where i come from only to engineering and science student, while in every math-majors-only-course stuff like "multiply by dx" or the things people do when abusing the substituion rule was forbidden.
>>7664347
Please explain in which way these are measures?
>>7664347
You mean in the context of lebesgue integration? Thats only kind of true but i think what
>>7664301
is talking about is the integration of differential forms. In that case dx is the 1-form (meaning differential form of degree one) that maps a vector of R^n to an alternating multilinearform of degree one over R^n. For suchs differential forms you can define an integral which matches the regular integral in the n=1 case, so you can interpret an integral over f(x) as the integral for the differential form of degree one f(x)dx. The qualities of differential forms give you some neat tricks so you can (relativly) easy proof strong results like stokes theorem for differential forms (which yiels the classical stokes theorem as an corrallary).
>>7664384
Forgot to add: I dont see how that exactly give you that tricks for differentiation though, considering the context is more integration.
>>7664369
I learned [math]\wedge[/math] in my proofs class. I use that symbol because it is the symbol of my math class and not my comp sci classes.
physics phd here
I only use Newtons dot notation when I can't be arsed to write df/dt, so it's only the time derivative
df/dx is useful because you can abuse it to get proper results
treat them like actual fractions
math fags will tear their hairs out, but at the end of the day: IT WORKS
look at derivations in theoretical physics
CS major here.
Euler's notation is best, and that's not surprising since he is the best mathematician of the lot.
>>7663504
Depends on the problem, faglord. Now go back sucking cocks and feeling smug about your calc fedora.
You need to in Calc III.
>>7663955
I particularly like how it gives the impression that you're dividing by zero and getting a meaningful answer. That's one of the reasons I've always liked Calc, along with adding an infinite number of infinitesimals: it's like it has a "do the impossible, row, row, fight the power" attitude.
>>7664577
https://en.wikipedia.org/wiki/Exterior_algebra
Just use & fool
>>7663525
just wait until summer for tolerable discussions
Does anyone know what the notation on the RHS is?
I've seen it used a few times
>>7664217
What if you wanted the change in y assuming X remains constant?
