Why do we write $df/dx$ instead of $df/dx(x)$?

Question

I was just thinking about how, i.e., if $f\colon\mathbb R\to\mathbb R$ is defined by $f(x) = x^2$, then it's customary to write $$ \frac{df}{dx} = 2x. $$ But since the derivative is itself a function from $\mathbb R$ to $\mathbb R$, shouldn't we write $$ \frac{df}{dx}(x) = 2x? $$ After all, we would typically write $f'(x) = 2x$, not just $f' = 2x$, right?

Answer 1

For your objection, I prefer: if $u = f(x)=x^2$, then $$ \frac{du}{dx} = x^2 \qquad \text{and}\qquad f'(x)=x^2 $$ Or even $$ \frac{d}{dx}\big(x^2) = 2x $$ Then (if you like) you can avoid both $$ \frac{df}{dx} = x^2\qquad\text{and}\qquad\frac{df}{dx}(x) = 2x $$