Would it make sense to use $\frac{df(x)}{dx}$ as a notation when differentiating a function?

Question

While I have never seen the above notation, it seems to make the most sense to me: you are considering the rate of change of $f(x)$ 'divided by' the rate of change of $x$. By contrast, writing $\frac{d}{dx}f(x)$ feels odd because $\frac{d}{dx}$ does not mean anything on its own. You need the $f(x)$ to make sense of what you are differentiating—the $d$ on top of the 'fraction' pertains to $f(x)$. Am I missing something?

Answer 1

Differentiation can be thought as applying linear operator to a function. I like to think about notation that confuses you, as applying that operator to each function.

While differentiating single function it may seem odd to make such notation, it becomes very intuitive, when you have to differentiate not only one function, but many:

$$\frac{d}{dx}\biggr[f(x)+g(x)+h(x)\biggl]$$

It shows that you have to apply linear operator to each of the function, while notation like: $$\frac{d\biggr(f(x)+g(x)+h(x)\biggl)}{dx}$$ may provide a little confusion, although it is still correct.

Answer 2

I find it odd that you say "I have never seen the above notation", $\frac{df(x)}{dx}$, while I have seen in just about every Calculus book I have looked at! You also say that $\frac{d}{dx}$ does not "mean anything on its own". Of course it does- it is the differential operator and has as much meaning as talking about the "sine function" rather than "sin(x)". Finally you say "you are considering the rate of change of (x) divided by' the rate of change of x". No, you are not. $\frac{df}{dx}$ is the limit of the average "rate of change of f(x) divided by the rate of change of x". The limit concept is crucial here.