Why is dy/dx sometimes written as a fraction in mathematics?

[hangover]

My high school teacher always says that the (dy/dx) should not be interpreted as "dy" divided by "dx". (dy/dx) is a symbol meaning the derivative.


However i often observed that mathematicians and physicists wrote such things as:

dy=3dx
instead of dy/dx=3

and even----(dy=3dx), divided by dt ,means (dy/dt=3dx/dt)
??

Do they break the convention or my teacher is wrong ?

 

[rasmhop]

Your teacher is right.

However:
When solving integrals some people like to use the notation dy=3dx to mean dy/dx=3 because this helps intuition, but in reality it's sloppy notation and really what they mean is dy/dx=3. The reason for this notation is that in some elementary ways this works out nicely. Some mathematicians allow this abuse, but others are adamant that it has no place.

Also later on you will encounter differential forms which gives meaning to expressions of the type dy, but that's a long way away* and until then dy has no independent meaning.

* At least not until you've been in college for some time.

 

[HallsofIvy]

My high school teacher always says that the (dy/dx) should not be interpreted as "dy" divided by "dx". (dy/dx) is a symbol meaning the derivative.


However i often observed that mathematicians and physicists wrote such things as:

dy=3dx
instead of dy/dx=3

and even----(dy=3dx), divided by dt ,means (dy/dt=3dx/dt)
??

Do they break the convention or my teacher is wrong ?

Well, they are playing a little fast and loose with it! (And your teacher is not wrong.)(And the fact that dy/dx is not a fraction is NOT a "convention".)

dy/dx is NOT a fraction- it is the limit of a fraction. And that means that, as long as you are careful, you "treat it like a fraction". For example, the chain rule: dy/dx= (dy/du)(du/dx) is true but not because you can "cancel" the du terms. It is true because you can go back before the limit, use the fact that the "difference quotients" are fractions to cancel the differences corresponding to "du" and then take the limit.

All formulas like dx/dy= 1/(dy/dx) that "look like" fraction properties can be proved that way. That is: dy/dx is NOT a fraction but it can be treated like one.

In order to be able to use the fraction properties without always making that "caveat", mathemticians define "differentials" (here we are talking in one dimension but they really become important in differential geometry). We take "dx" as purely "symbolic" and then define "dy= f'(x) dx". Now we can say "f'(x)= dy/dx" and have "dx" and "dy" defined separately so that dy/dx really is a fraction. Of course, now, "dy/dx" as a fraction of differentials is NOT the same as the derivative, "dy/dx", but we allow the "abuse of notation" because it leads to correct results rather than causeing confusion.

Of course, we have to be careful how we use it- unlike "regular" fractions, we cannot use dx and dy separately. We cannot have an equation that has a "dx" without a "dy" in away that one can be divide by the other, or without an integral.

 

[g_edgar]

In even more advanced mathematics ... differential topology, say ... you talk about "differentials" and really do write dy = 3 dx and such things.

 

[Tac-Tics]

dy/dx and its kin are notational abuse.

The problem was the derivative is an operation on functions. But the formal notion and proper notation for a function didn't come into existence until later in history.

The trick is that most of the time, the dy/dx notation can be treated like a fraction and still give you the right answer. Not always, but most of the time. This means you really have to be careful and stick to the methods you're taught in class.

Later on, in real analysis, you learn how calculus *really* works.