Question regarding the derivative terminalogy and wording

[Mr Davis 97]

When doing calculus, we typically say that we "take the derivative of a function $f(x)$." However, rigorously, $f(x)$ is not a function but rather the value of the function $f$ evaluated at $x$. Thus, in order for this wording to be correct shouldn't we have to write something like $\displaystyle \frac{d}{dx}(x \mapsto f(x) )$ instead of $\displaystyle \frac{d}{dx}f(x)$, as we normally do? Something about this mix up of words and notation bothers me.

 

[andrewkirk]

You're right. Out of the several notations available for derivatives, the $\frac{d}{dx}$ notation is the least clear and the most subject to misuse and subsequent confusion. I think the 'prime' notation is better, as $f'$ is a function that is the derivative of $f$, so that $f'(x)$ is $f'$ evaluated at $x$.

One advantage of the $\frac{d}{dx}$ notation though is that it has a clearer link to the definition of the derivative as the limit of a ratio.

$\frac{d}{dx}f(x)$ can be OK if you infer the existence of implied parentheses in the right place, ie $\left(\frac{d}{dx}f\right)(x)$. Another way to write it, following the same principle, is $\frac{df}{dx}(x)$.

The muddle can get much worse when one does partial derivatives. It's possible to write horribly ambiguous and misleading formulas using fairly commonly-used notation.