Understanding dy/dx in Calculus

[david90]

I don't get the concept of the notation dy/dx. Sometimes my physics teacher puts dx in the back of an equation and cross multiply with other number. Does this mean dx can be multiply like numbers? I just don't get the overall concept of dy/dx.

I'm in 3rd year calculus and I understand that derivative is the instantaneous rate of change but how does it relates to the notation?

 

[abhishek]

I hope I don't confuse you more (I'm not entirely certain myself).

The notation dy/dx comes from Leibniz's notation of derivatives. Originally, he used the Greek small delta (δ) which looks a bit like a 'd'. Anyway, the d or delta represents 'a small change in'. So dy/dx means 'the small change in y, with relation to x'.

Now the dy/dx notation has one useful property to it, and that is that it can behave as a fraction. Consider the Chain Rule:

dy/dx = dy/du * du/dx

The du's can cancel as though they were in fractions, to give you dy/dx again.

This becomes more pointed with differential equations and the relation with integration, where you can 'move' the dx to be able to integrate a DE.

I think that should about cover the very basic info about dy/dx notation... you'll probably get a better answer from a real mathematician...

Good luck.

 

[HallsofIvy]

One of the things your textbook should make clear (and you should ask your teacher about) is the distinction between "derivative" and "differential".

The derivative (represented y'(Newton's notation) or dy/dx (Leibniz' notation)) is NOT defined as a fraction. Strictly speaking it is incorrect to separate the "dy" from the "dx" in the derivative.

HOWEVER! dy/dx IS defined as the "limit of a fraction". One can prove "fraction-like" properties (chain rule: dy/dz= (dy/dx)(dx/dz) for example) by going back before the limit, cancelling parts of fractions, and then taking the limit. That is, we can always TREAT a derivative like a fraction. To take advantage of that, we define "dx" purely symbolically and then define dy by "dy= f'(x) dx". Given that definition, dy/dx DOES represent a fraction! Since dx is only defined symbolically, you should never have a dx in an equation without a corresponding dy (or vice-versa) unless it is in an integral (which effectively removes the derivative).

If you are wondering what I mean by "define symbolically", well, we can define differentials precisely in "differential forms" but that is beyond calculus so just "think" of it as symbolic.
You might want to check out Lethe's thread "differential forms" under the differential equation forum.