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When doing integration and calculus why can we manipulate $dy/dx$ as if it was a fraction. I heard this only works when working in one dimension or something? And aren't $dy$ and $dx$ supposed to be quantities denoting a very small change of value in x and y. Could someone explain?

Tom Xia
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  • You can extend the intuition of the derivative operation by treating $dy/dx$ as a fraction in any number of dimensions, however this is not really a very formal definition. – PC1 Jul 04 '23 at 01:34
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    In standard analysis—which is the basis for almost every calculus course—I would consider “treating $dy/dx$ as a fraction” to be a useful mnemonic for certain theorems, but not an actual mathematical tool. In other words, on you have a theorem that says you can manipulate derivatives in a certain way that looks like treating them as fractions, you can use that operation on fractions to remember the theorem, but don’t apply any new operations on fractions to derivatives until you have confirmation that they work. – David K Jul 04 '23 at 01:42
  • It is a shorthand for the chain rule, see here. – Ninad Munshi Jul 04 '23 at 02:04

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