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

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The discussion centers on the chain rule in calculus, specifically the expression (dy/dx) = (dy/du)(du/dx). Participants clarify that while dy/du and du/dx resemble quotients, they are not legitimate quotients and cannot be canceled out. The consensus is that canceling du's is incorrect because they represent derivatives rather than independent entities. Understanding this distinction is crucial for correctly applying the chain rule in calculus.

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

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