dy/dx as a fraction

[b^3]

I think this has been answered before but I can't seem to find the thread.

Anyway, dy/dx is not a fraction but rather an expression for the derivative of y in respect to x correct? It does obey fraction laws but is it acutally correct to say something like du=2dx? Are we allowed to write in in our formal working out on the exam or avoid writing it and just move on to the next step?

Sorry guys and girls if it has been answered before (I'm pretty sure it has though)

[Lasercookie]

There'll be someone who can explain this better than me, so I won't even try and make an attempt - I'd probably get things pretty wrong. I've just done a bit of searching:

I believe this was the thread you were talking about (I searched "TrueTears dy/dx fraction" - assumed that if anyone discussed this before it'd be TT lol): Re: Gloamy's Thread of Kweshchuns

Simple explanation: http://mathforum.org/library/drmath/view/65462.html

http://mathoverflow.net/questions/73492/how-misleading-is-it-to-regard-fracdydx-as-a-fraction (also links to a lot of relevant threads - haven't put in the effort to understand what was said yet)

http://www.physicsforums.com/showthread.php?t=401409

http://books.google.com/books/about/Ordinary_differential_equations.html?id=29utVed7QMIC - Apparently early on in that book there's a good "elementary explanation of differentials" (it appears that dy and dx are differentials - I don't know what that means lol). Most of it seems to be in the preview, I'm taking a guess that it might be "Lesson 3"

[paulsterio]

dy/dx is not a fraction, but is the limit as the change in x approaches zero of the change in y over the change in x

i can't be bothered with latex, but basically lim delta-x ->0 of delta-y/delta-x

that's why dy/dx behaves like a fraction, but the second derivative does not (:

and to add to your question, its legit to write dy = 5dx for example, there's nothing wrong with that

[Mao]

http://en.wikipedia.org/wiki/Exact_differential

I treat it as a 'fraction' when I need to change variables. This exact relationship between differentials is one of the ways we can perform u-substitution.

But in general, it is not a 'fraction'.

[paulsterio]

But in general, it is not a 'fraction'.

i agree, it's not a fraction
but for the sake of high school maths - it helps with simple arithmetic to say that it's a fraction
there's an example here on AN about the two ways you can approach the substitution
the dx = ......du way and the du/dx = .... and we try to find something indicative of that in the integrand and sub the whole du/dx in
and the dx = .....du way was shown to be far quicker (important in an exam i guess)

but unless you're really going to do maths or physics in uni
anything to do with partial derivatives - ie. lagrangian mechanics for the physics kids
it won't hurt to say dy/dx behaves like a fraction

[b^3]

OK thanks for that guys. 

[Mao]

But in general, it is not a 'fraction'.

i agree, it's not a fraction
but for the sake of high school maths - it helps with simple arithmetic to say that it's a fraction
there's an example here on AN about the two ways you can approach the substitution
the dx = ......du way and the du/dx = .... and we try to find something indicative of that in the integrand and sub the whole du/dx in
and the dx = .....du way was shown to be far quicker (important in an exam i guess)

but unless you're really going to do maths or physics in uni
anything to do with partial derivatives - ie. lagrangian mechanics for the physics kids
it won't hurt to say dy/dx behaves like a fraction

well, a better way to look at it is examining what a differential is, and the shortcomings of different notations.

What is not taught in the classroom is that and are not part of the derivative. In this sense, these are non-standard interpretation of a first order derivative, especially considering the second order interpretation does not exist. i.e. has the interpretation of the quotient of two infinitesimals. where as the second derivative does not have a similar interpretation. Non-standard analysis (the mathematics of infinitesimal quantities) formulates the concept of infinitesimals via an important axiom. Using Liebniz's notation,  , or using Lagrange's notation, . That is, we relate the exact differential to an infinitesimal quantity via the derivative.

This is the problem with notation. Because appears both as a differential, as well as part of Liebniz's notation, people can confuse the two together and treat the derivative as a fraction. A better way to think of Liebniz'z notation is to always think of it as a differential operator. I.e. . In this sense, it is just a special operation, like sin, cos and log. Euler's notation , where achieves the same thing, and is used in higher level mathematics. The best apprach to understanding differentials is to use Lagrange's notation, , this eliminates any ambiguities.

Note that in the integral is a differential, and it is defined from the exact differential relationship .

This allow us to do many things, such as u-substitution and separation of variables, but we NEVER split up the fraction. Whenever we do this, we have actually introduced new differentials. Pay close attention to the following example:

Solve .

What you think you are doing:

, splitting up the fraction, , thus

What actually happened, and the steps that you've skipped:

. Given the exact differential relationship , we substitute the derivative. Thus , thus

The difference is subtle, but it's there. And it keeps the world going around. Obviously in VCE you don't need to be this rigorous, but a proper understanding is crucial at the university level, as equations are no longer in terms of one variable, and the definition of a differential is required so you don't accidentally miss a whole variable.