Are all notation equal by derivatives?

Question

Here I read that

Suppose you have a general function: y = f(x). All of the following notations can be read as "the derivative of y with respect to x" or less formally, "the derivative of the function": f'(x), f', y', df/dx, dy/dx, d/dx[f(x)].

Still if I want to describe the value of the derivative at a certain point I write f'(3) for example and not df(3)/d3. I am not sure if the latter is syntactically valid, because I have never seen it. Is it possible with all of these notations to describe the value of the derivative for a certain x? If so, then how? If not, then why? Are there abstract and concrete notations in math?

Answer 1

All of those notations for the derivative: $$f'(x),\quad\frac{df}{dx},\quad\frac{d}{dx}f(x),\quad\frac{df}{dx}(x),\quad\frac{df(x)}{dx}$$ are equivalent. However, if we replace $x$ by a constant, some of these notations no longer make much sense. In Leibniz notation (all of the latter four, i.e. those with a $\frac{d}{dx}$), we are thinking of the derivative $\frac{d}{dx}$ as an operator acting on the function $f(x)$ (or $f$). So if we were to write $$\frac{d}{dx}f(3),$$ this seems to imply that the differentiation operator is acting on the constant value $f(3)$. But then we are just differentiating a constant, which gives zero. On the other hand, the notation $f'(3)$ is interpreted as plugging in the value of $x=3$ into the function $f'(x)$, and the differentiation operator has already acted on $f(x)$ before plugging in the value. In this case $f'(3)$ is not zero (in general). If you want to achieve the effect of plugging in a constant value with Leibniz notation, we usually use $$\left.\frac{df}{dx}\right\vert_{x=3}\quad\text{or}\quad\left(\frac{df}{dx}\right)(3),$$ though I should warn you that the latter is rather uncommon and nonstandard. In either case, notation makes it clear that the differentiation is done first, and the plugging in of the constant value done second.

Answer 2

All of those not on the ends are equal, as they all denote the same thing. The last one, I've never seen. The first one is "close" - I'm going to get to that.

However, there is actually a very good question you ask here - namely, if we have

$$\frac{df}{dx}$$

how can we write, with this notation, what the value at $x = 3$ is? And the "best" answer to this is that we have to make a clear distinction between three different things:

1. the meaning of the symbol "$f(x)$",
2. the meaning of the symbol "$f$", and
3. what kind of "thing", "differentiation", is.

And this stuff typically isn't touched on as well as it could be, and when it needs to be. The symbol $f(x)$ is not a function, even though it often gets abused by calling it such. Rather, what it is is the value of the function at $x$, and if $x$ is a variable that hasn't been assigned a value yet, this value is also unspecified (except that it must be within the range of the function). That is, $f(x)$ is a symbol for a (possibly unspecified) number.

What $f$ is, on the other hand, is the function itself: the mathematical object that you can plug numbers into to get other numbers out. Thus, you should, ideally, never say "the function $f(x)$", when what you really mean is "the function $f$", according to the above understanding.

Now, for the last part. What "differentation" is, is what is called an operator, or a higher-order function: it is a function that takes in other functions as inputs and pops out functions. Such things are typically denoted by prefixing their symbol against the function symbol: hence here, you should have $\frac{d}{dx} f$, which is shortened to $\frac{df}{dx}$, and this forms a combined function symbol.

Hence, if you want the value at $x = 3$, say, I'd argue you should "best" write

$$\left[\frac{df}{dx}\right](3)$$

which means to evaluate the function denoted by the symbol $\frac{df}{dx}$ at 3. And this, thus, generalizes likewise to higher derivatives, e.g.

$$\left[\frac{d^2 f}{dx^2}\right](3)$$

and so forth.

Of course, if you think about this even more, technically we should not ever write something like

$$\frac{df}{dx} + x$$

even though this is very common, e.g. in writing a differential equation, and you can "get away with it" in that people will generally know what you mean. This is because here, $x$ is a number, and $\frac{df}{dx}$ is a function. To be "absolutely" correct, you will have two choices: either demote $\frac{df}{dx}$ to a number, which will make it

$$\left[\frac{df}{dx}\right](x) + x$$

or, to promote $x$ to a function, in which case, if you don't want to give another function symbol, you have to use the anonymous function notation

$$\frac{df}{dx} + \left(x \mapsto x\right)$$

Either of these would be a fully-correct mathematical expression (though keep in mind, they actually have a different, but related, meaning each! What is it?) But since they are a bit ungainly, we accept the "wrong" notation as a shorthand: it's called "abuse of notation". And there isn't anything wrong with that, as long as you clearly understand what is going on, which you can check by trying to mentally translate it into the strictly proper, if not typically used, notation. Just like learning a foreign human natural language, where that in typical speech we'll never always follow the rules of grammar perfectly, but we always try to learn the ideal forms first, when learning the mathematical language we should keep in mind the same principle.