Is this notation a little bit ambigious?

Question

We are all used to seeing $y=f(x)$ where we wish to plot the function $f$ on the $xy$ plane. We can differentiate both sides, $\frac{d}{dx}(y) = \frac{d}{dx}(f(x))$ to get the good old $\frac{dy}{dx} = f'(x)$, now we know that $\frac{dy}{dx}$ and $f'(x)$ mean the same (the derivative of $f$), but also I have seen $\frac{df}{dx}$ written. First off, is this shorthand for $\frac{df(x)}{dx}$ and second of all, in the statement $y=f(x)$ we know $y$ is not a function, its just a variable that takes the value of the function $f$ at each point $x$. That means we can translate $\frac{dy}{dx}$ to mean "the rate of change of the variable $y$ as $x$ varies". And we can translate $\frac{df(x)}{dx}$ to mean the same, as $y$ and $f(x)$ mean the exact same thing, they are both ouputs of the function $f$ (and we are interested how fast/slow these outputs change as we vary the input $x$). Taking a look at $\frac{df}{dx}$ one might interpret it to mean, "the rate of change of the variable $f$ as $x$ varies" but that's wrong, its actually "the rate of change of the output of $f$ as $x$ varies). Why do mathematicans not just write $\frac{df(x)}{dx}$ so there is no chance someone interprets $f$ as a variable?

Answer 1

I agree with you. But, this abuse of notation (conflating a function $f$ with its value $f(x)$) appears all over mathematics, and usually it's innocent enough as there's no ambiguity. But you are correct to note the distinction.

If we're being careful, the derivative of a function is a function, so we probably ought to write something like $$ f' = \frac{dy}{dx} $$ and if we choose to evaluate at a particular $x=a$, say, $$ f'(a) = \frac{dy}{dx}\bigg|_{x=a}. $$ This is annoying because it requires an additional symbol when we intend for the $x$ in $f(x)$ to be generic anyway.

At the end of the day, context is everything. Notation works for you, not the other way around. When you write an expression, ask yourself is this a function or a value of the function (or something else).

Answer 2

Well, because $f$ is a variable, even when it’s the name of a function. It varies. It is simply a difference in perception and appearance, not a genuine mathematical difference.

You might also see $y(x)$. And when you really need to specify when a variable is dependent or independent, it’s important to be careful with all this. But this single variable distinction you are talking about does not really matter.