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In lower level courses, we are frequently told that we can 'cancel out' notation, even though this is not actually true. This false reasoning is then justified for the sake of simplicity. This is frequently done in various differential equations.

$\int \frac{ \partial x}{\partial t} \partial t = \int \partial x$ ?

I'm curious what the real reasoning behind this is. Why are we able to (incorrectly) think of it as 'cancelling' notation? What is the REAL reason behind why this occurs? I have been told that the theorem responsible for this is the chain rule but this is the limit of my understanding.

I would appreciate elaborate and clear reasoning. I would also be thankful for an example to assist in illustrating any reasoning. Thank you.

The Pointer
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    A specific example might be more helpful, since as stated the question is probably too broad to answer. Also you might want to look at all of the questions about "when can a derivative be treated as a fraction" or things along those lines. – Chill2Macht Oct 11 '16 at 09:07
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    Also, read up on "integration by substitution". Many sources on that topic give a proof that has nothing to do with 'cancelling fractions'. – Arthur Oct 11 '16 at 09:21
  • Thanks. For anyone else interested, the following question seems to be a good reference: http://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio – The Pointer Oct 11 '16 at 09:42

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