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This has always been a source of confusion for me that everyone seems to play around with dx and dy as if they were variables while in many sources it was stated that they are purely symbolic. For example, in the integral, dx is purely symbolic. If I'm not misunderstanding, dx is in fact part of the integral symbol. Yet all my professors manipulate these "symbols" endlessly when performing integration. Even worse is when the integral is set up for volumes and dx seems to play the part of the missing dimension (infinitesimal width, etc).

Now my rough understanding is this: although dy/dx is just a "symbol" for derivative, fraction like properties can be proven for it, justifying our manipulation of these symbols as fractions. Similarly for the integral, substitution by parts allows us to prove fraction like properties as if the dx were being multiplied into the integrand. Yet when I did further research I found a wikipedia page that states "does not always behave exactly like a fraction (e.g. dx/dy is not always equal to 1/(dy/dx))". Ironically, this was under the page for "Abuse of Notation". I'm not sure whether wikipedia is correct or not but my question is then:

What is the justification for these fraction like manipulations (I would love to see something like a proof, but if that is too long or complicated then a simple verbal comment would be just as good) and when are (if there indeed are) there exceptions?