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.