My Calc teacher told us that $dy\over dx$ does not mean $dy$ divided by $dx$. However, it seems like writing differential form involves only multiplying both sides by $dx$. Why is this not just division?
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1https://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio ? – For the love of maths Dec 28 '18 at 17:35
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This is a remnant of the origins of Calculus. Essentially, $dx$ means a very tiny change in the $x$-value, while $dy$ is the small change between the old $y$ value and the new. $dx \over dy$ is the ratio of the changes. In many ways, this is division. However, division by zero is invalid, and $dx$ and $dy$ should go to $0$ for a continuous function.
William Grannis
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