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.

If I give my answer from an eye of physicist ,then you may think of following-
For a particle moving along $x$-axis with variable velocity, We define instantaneous velocity $v$ of an object as rate of change of x-coordinate of particle at that instant and since we define "rate of change", therefore it must be equal to total change divided by time taken to bring that change. since we have to calculate the instantaneous velocity. we assume instant to mean " an infinitesimally short interval of time for which particle can be assumed to be moving with constant velocity and denote this infinitesimal time interval by $dt$. Now particle cannot go more than an infinitesimal distance $dx$ in an infinitesimally small time. therefore we define instantaneous velocity as
$v=\frac{dx}{dt}$  i.e. as ratio of two infinitesimal changes.
This also helps us to get correct units for velocity since for change in position it will be $m$ and for change in time it will be $s$.
While defining pressure at a point, acceleration, momentum, electric current through a cross section etc. we assume ratio of two infinitesimal quantities.
So I think for practical purposes you may assume it to be a ratio but what actually it is has been well clarified in other answers.
Also from my knowledge of mathematics, when I learnt about differentiating a function also provides slope of tangent, I was told that Leibniz assumed the smooth curve of some function to be made of infinite number infinitesimally small lines joined together ,extending any one of them gives tangent to curve and the slope of that infinitesimally small line = $\frac{dy}{dx}$ =  slope of tangent that we get on extending that line.
Even I learnt that we would need a magnifying glass of "infinite zoom in power" to see those lines which may not be possible in real.
