Can the du's be cancelled out in the derivative (dy/dx)?

[wajed]

"
(dy/dx) = (dy/du)(du/dx)

Caution: Notice that if (dy/du) and (du/dx) were quotients rather than expressions for derivatives, we could cancel the du`s and make (5) into an identity. But we stree that du has not been defined as an entity, and consequently it is not legitimate to cancel the du`s. Nonetheless, the resemblance between (5) and an algebraic identity makes it easy to remember.
"

so, he`s saying that I can`t cancel out the du`s, because they are not quotients, but expressions for derivative..
but I think that they can be canceled out!? is that true?
and yes they are expressions for derivative, but also they are quotients..
and I understand that we need them, and it would be meaningless to cancel them..
its just that I think he`s giving me wrong information when he said that I "can`t cancel them out" because "they are expressions for derivatives"

Am I wrong?

 

[owlpride]

That's the chain rule. It is a legitimate identity, but it is not quite as immediately obvious as some may think.

 

[matt grime]

You should not think of them as quotients.

The derivative dy/dx is the limit of quotients. That is *not* the same thing as being a quotient. In some cases identities can be remembered because things cancel as if it were legitimate to separate the dy and dx. You should *not* think that you can separate them out at this stage.