Why don't pure mathematicians like to split derivatives?

[KingBigness]

I remembered the other day that a while back my teacher (can't remember what topic, think it was implicit differentiation) had something like dy/dx and said to us that you could times both sides of the quation by dx as long as you don't tell any pure mathematician.

I was wondering why pure mathematicians don't like this if it gives the correct answer.

Sorry for the dodgy post I just can't remember all the details of what was going on as it was a few months ago.

Hope it isn't to unclear =S

 

[micromass]

Aaaargh. Your teacher is correct: you shouldn't have told me that.

The reason is simply this: dx does not exist. It stands for nothing. So what are you multiplying with?? Something that doesn't exist.

$$\frac{dy}{dx}$$

is simply a notation, it is not a fraction

And it doesn't always give the right answer. For example the formula

$$\frac{\partial u}{\partial r}=\frac{\partial u}{\partial x}\frac{\partial x}{\partial r}+\frac{\partial u}{\partial y}\frac{\partial y}{\partial r}$$

and this shows that we cannot simply scratch the $\partial x$ and $\partial y$ in the numerator.

Everybody who says that you can "just multiply by dx" is lying.

 

[KingBigness]

Ok I see I thought as much.
So is there a reason it worked in his case? Or did it really not change anything it just "simpfied" the notation.

 

[KingBigness]

Ps, sorry to cause you so much distress =P

 

[micromass]

Ok I see I thought as much.
So is there a reason it worked in his case? Or did it really not change anything it just "simpfied" the notation.

There is a deeper reason why it works. The idea is to give dx a meaning. This is done in nonstandard calculus and differential geometry. But these topics are usually not treated in a calculus course.

So your teacher is not completely wrong here. You can just multiply by dx and get the right answer. But he shouldn't have said that without explaining all the deeper theory. He should at least have mentioned that it isn't as trivial as you think. The last thing that I want you to think is that dx is some number and that $\frac{dy}{dx}$ is a fraction and that everything cancels out nicely. This is fundamentally false.

Ps, sorry to cause you so much distress =P

It's your teachers fault, not yours

 

[KingBigness]

Thanks for that explanation.
I understand it's not a number that's why I am asking this because I know he is wrong just wanted to know why. It unsettles me when people say, this works as long as you don't ask someone who is good at it
Haha

You seem like you know what you are talking about.
I have a partial derivative question I can't do and I have put it in the homework section to no avail.

Do you mind checking it out? It's called partial derivative proof, or I can pm you the question

Thank you again!

 

[KingBigness]

Sorry to revive a dead thread, I just thought you would be the best to answer my question as it follows along from my previous one.

You say you can't split dx because it 'doesn't exist'.

I have just started errors with derivatives. It says that the ratio of the two derivatives is actually the derivative of the function.

f'(x)=\frac{dy}{dx}

and the relationship between the two differentials can be given by

dy=f'(x)dx

Is this not 'splitting' the derivative?