Question

Is there any reason why dy/dx isn't a fraction? I always treat it as such and I have never had a problem.

Answer 1

Examples of using it as a fraction:

John runs twice as fast as Karl, Karl runs 3 times as fast as Lynn, how many times faster does John run than Lynn? Obviously 6 times right? \[\Large \frac{dJ}{dK}=2\]\[\Large \frac{dK}{dL}=3\]\[\Large \frac{dJ}{dK} \frac{dK}{dL}=\frac{dJ}{dL}=2*3=6\]
Just the chain rule basically.

Suppose we're integrating, \[\Large \int\limits \frac{x dx}{x^2+1}\]\[\Large x^2+1=u\]\[\Large 2x=\frac{du}{dx}\]\[\Large du/2=xdx\]

Yeah, I don't get it.

Answer 2

well who says they aren't fractions? :>
the entire bloody world of ordinary differential equations sort of depends on us being able to separate the dy/dx stuff into dy's and dx's so that we can integrate...`

Answer 3

maybe calculus newbies are just discouraged from treating dy/dx as a fraction until they have a more thorough understanding of what the individual differentials dy and dx actually mean... no biggie, right? ^^

Answer 4

By complete coincidence I was just told by someone a few minutes ago in this question! XD

http://openstudy.com/study#/updates/53c27b6ee4b09e08a1c7c2be

Answer 5

\[\Delta y = f'(x) \Delta x\]

Answer 6

i guess things won't break if we stick to single variable; the fraction business won't work if you try to extend it to chain rule in two or more variables

Answer 7

Hmm, I guess so\[\Large \frac{du}{dt}=\frac{\partial u}{\partial x}\frac{dx}{dt}+\frac{\partial u}{\partial y}\frac{dy}{dt}\]

Answer 8

\[\frac{ dy }{ dx }=\frac{ dy }{ du }\frac{ du }{ dx}\]

Answer 9

I remember learning the chain rule like this, but I had to relearn it as a composition function, easier to explain it as a composition function

Answer 10

It has been so long since I did it in this fashion. I am probably wrong.

Answer 11

My professor told , we can treat it as a fraction , It is called seperation of variables.
It is often used in proving basic kinematic equations.

If you think it logically,
It is nothing but
a small small small change in x over a small small small change in y

However small it is a fraction because it has a numerator and a denominator

Answer 12

He said it is also called the fourier method

Answer 13

leibniz is the culprit for this, not fourier lol

Answer 14

culprit for?

Answer 15

@Kainui

Answer 16

I said it is called the fourier method , (seperation of variables) , he told me

Answer 17

Haha, well I think for the most part you won't run into problems as treating these infinitesimals as being parts of fractions, but occasionally you might I suppose. By culprit all ganeshie means is that Leibniz is who came up with the dy/dx notation while Newton used the dot notation. These are generally the three ways people denote derivatives of single valued functions. \[\Large f'(x)=\frac{dy}{dx}=\dot y\]

There's a 4th way that I prefer for partial differential equations, all that it does is look like this: \[\Large \frac{\partial u}{\partial x}=u_x\] Although you haven't gotten to partial derivatives yet so don't worry about this too much.

Answer 18

Yes, separation of variables is the fourier method, and it does treat the infinitesimals as quantities you can move around like fractions.

Answer 19

Isn't that called parametric

Answer 20

chain rule?

Answer 21

Parametric in my experience refers to making a vector equation.

So you can graph y=x^2 as being y dependent on x or you can have both y and x be dependent on some single variable like t and we can rewrite this as:

u=<x, y> = <t, t^2>

So we have \[\Large u(x(t),y(t))\]

So u is a function of x and y, while x and y are both functions of some parameter t. Generally you can think of this as time and the path of a particle in space at any time kind of thing.

Answer 22

sorry i mean partial derivatives

Answer 23

Oh yeah, I thought I said that haha. XD

Answer 24

that's notation for that

Answer 25

oh yes hehe

Answer 26

=D

Answer 27

lol never dealt with it as not a fraction ( even in proving ) !
since we have dy with responding to dx , if its a fraction then
y'' is also a fraction mmm although i can't say its a fraction since
its a consept but we deal with it like a fraction xD
but we when define a fraction we say a/b such that b\(\neq 0\)
but we dnt do that with dy/dx mmm i mean whats the meaning of dx=0 ?!
means no interval mmm idk lol need to think more deeply

Answer 28

Actually when I think of dx, I essentially think of it as being zero. It makes it understandable to me at least that if we look at

dy/dx we're really looking at 0/0 which sort of cancel each other out to give a normal number out. And then especially for integrals.

An integral to me is like an infinite sum of things, with each of the members multiplied by zero. So adding up an infinite number of zeroes sort of cancel each other out to leave you with a regular old number again, similar to 0*infinity or 0/0.

To me it seems that really we're allowed to use infinity and divide by zero as long as we're doing it in the right context. Limiting is the process by which we evaluate division of 0 by 0. But that's just sort of my heuristic approach to understanding it that seems to work.

Answer 29

yep , one reason to not consider it as normal fraction