# In Wikipedia, it is said that$$\mathrm dy=\frac{\mathrm 1. Aug 25, 2011 ### dalcde In Wikipedia, it is said that [tex]\mathrm dy=\frac{\mathrm dy}{\mathrm dx}\mathrm dx.$$
Can we divide both sides by $\mathrm dx$ and say that the derivative is $\mathrm dy$ divided by $\mathrm dx$?

2. Aug 25, 2011

### LCKurtz

Re: Differentials

Yes. That is the usual convention. I think it is less confusing for a calculus student to introduce differentials using the prime notation like this: Given y = f(x) and y' = f'(x), if dx is a small nonzero change in x we define the symbol dy = f'(x) dx. dy can be thought of the change in the y value of the tangent line or the approximate change in y if dx is small. In any case, if you divide both sides by dx you have dy/dx = f'(x).

3. Aug 25, 2011

### Dr. Seafood

Re: Differentials

It looks like ordinary division of numbers, but $\mathrm dx$ and $\mathrm dy$ are not ordinary numbers. However, we manipulate them symbolically in a way that appears like they are real numbers, for the sake of intuition. But we can do this without loss of precision! A good demonstration of it is using the substitution rule for integrals:

$$\int f'(g(x))g'(x) \mathrm dx$$

Substituting u = g(x), we get, by the chain rule for derivatives,

$$\int f'(u) \frac{\mathrm du}{\mathrm dx} {\mathrm dx} = f(u) + C = \int f'(u) \mathrm du$$

The last equality is given by the definition of antiderivative. Though we didn't actually "cancel out" the differentials like fractions, it does turn out that we can safely write $\frac{\mathrm du}{\mathrm dx} {\mathrm dx} = \mathrm du$, by the chain of equalities. Note that the justification for this, however, has nothing to do with fractions.

Last edited: Aug 25, 2011
4. Aug 25, 2011

### dalcde

Re: Differentials

I mean, if we define dy and dx like that, then is it totally correct to say that dy/dx is dy divided by dx?

5. Aug 25, 2011

### Dr. Seafood

Re: Differentials

Don't you really have to define "divide" (namely, "division") in order to do that? But "divide" typically refers to an operation involving numbers -- and differentials aren't numbers.

6. Aug 25, 2011

### dalcde

Re: Differentials

Well, we can define division for differentials, why not?

7. Aug 25, 2011

### Dr. Seafood

Re: Differentials

Alright, then wouldn't we be interested in defining what a differential is first, so we can define operations on them?

I think in the elementary, traditional sense, "division" here doesn't really work. I've always thought $\frac{\mathrm dy}{\mathrm dx}$ was cool-looking, but a bit notationally abusive.

8. Aug 26, 2011

### Rodrae

Re: Differentials

You might do that.. Sometimes there are things that is written in a book, etc. that is really confusing to understand. Maybe a simple explanation might be helpful when you are just starting calculus rather than those with lots of formula. Like those that I red before about the law of derivatives that when i read it. It is so confusing so i just make my own shortcut formula rather do those.

9. Aug 26, 2011

### hunt_mat

Re: Differentials

Hmmm, differentials are really part of of exterior algebra which is a very important tool for differential geometry. So I think you should look into exterior algebra for a firm understanding of a differential.

10. Aug 26, 2011

### dalcde

Re: Differentials

Do you have any resources for studying exterior algebra? I'd prefer some online resources since I can't gain access to any bookstores or libraries that have some serious mathematics books (I'm still at junior high school).

11. Aug 26, 2011

### hunt_mat

Re: Differentials

Any set of notes on elementary differential geometry will do but google is your friend...

12. Aug 26, 2011

### dalcde

Re: Differentials

Thanks. Just needed to know what to search for (I tried to search for exterior algebra but every result turned out very difficult.