Derivative Symbols

Question

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?

Answer 1

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.