Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? In Thomas's Calculus (11th edition), it is mentioned (Section 3.8 pg 225) that the derivative $dy/dx$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $dy = f'(x) \, dx$ we are able to plug in values for $dx$ and calculate a $dy$ (differential)? Then, if we rearrange we get $dy/dx=f'(x)$, and so $dy/dx$ can be seen as a ratio of $dy$ and $dx$. I wonder if the author says this because $dx$ is an independent variable, and $dy$ is a dependent variable, and for $dy/dx$ to be a ratio both variables need to be independent.

It is incorrect to call it a ratio but it's heuristically handy to visualize it as a ratio where the $\frac{l}{\delta} $ talks about the times a certain $l>0$ can be divided by saying an exceedingly small $\delta>0$ such that there is an exceedingly small remainder. Now if even $l$ is exceedingly small then the relative difference of the two changes can be measured using this "ratio". The idea of limits suggests that these two values $l$ and $\delta$, are themselves changing or $l$ changes with $\delta$ and the sole reason this can't be a ratio is that ratios are defined for constant fractions whilst $\frac{d}{dx} $ is a changing fraction with no concept of a remainder.
