Why is this the notation for taking derivatives?

Question

For a function like $y = x^2$, the derivative is $dy = 2x\ dx$ and its formally rewritten as $\frac{dy}{dx} = 2x$. This makes sense to me logically because its saying the change in y over the change in x is 2x. How come when my teacher tells us to take the derivative of a function he will write it like this: $[\frac{d}{dx}] y=x^2$? To me, I see this as "change over change of x". I know this notation means take the derivative of the function, but what does $\frac{d}{dx}$ mean exactly?

Answer 1

The notation has no meaning beyond "take the derivative with respect to $x$" and should not be interpreted literally. The idea is that formally if you "multiply" $\frac{d}{dx}$ by $y$ you get $\frac{dy}{dx}$, but this is is really just a "notational joke" since it does not really make sense to separate the $d$ from the $y$ like this. You should think of $\frac{d}{dx}$ as a single indecomposable unit that just means "take the derivative with respect to $x$".