How to express that dx is a single variable?

Solution 1:

As people in the comments have pointed out, what you are writing down doesn't make much sense. Admittedly, there are numerous ways to understand the derivative (e.g. as rate of change, as an operator/linear map, etc.). Symbolically, writing the "$dx$" in the 'denominator' of $\frac{d}{dx}$ is not to interpret $dx$ as a variable. This is just how the derivative is written. What you might be getting confused about is the fact that the $dx$ in the bottom of $\frac{d}{dx}$ denotes the derivative of a function with respect to the single variable $x$.

In general, we could have a function $f(x,y)$ which depends on two variables $x$ and $y$, but we might only be interested in its derivative with respect to $x$. In which case, $\frac{d}{dx}f$ (in this case, we usually use the slightly different notation $\frac{\partial }{\partial x}f$ for partial derivative) denotes the derivative with respect to the single variable $x$ (despite $f$ depending on two variables $x$ and $y$).

In summary, $dx$ is not a single variable (whereas $x$ is a single variable). In fact, what $dx$ really is depends a lot at the level of mathematical sophistication you are at (we usually think of $dx$ as something called a $1$-form, $d$ being something called the exterior derivative). But for our intents and purposes, we should just think of $dx$ appearing in the derivative $\frac{d}{dx}$ as denoting taking a derivative with respect to $x$.