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The derivative of a function is often defined as $f'$ and $f'(x)$. So which one is it? $f'(x)$ is the output of the function $f'$, so why do I see people using $f'$ and $f'(x)$ interchangeably to refer to the derivative of a function?

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  • $\begingroup$ By the way, check the first bullet point here $\ddot \smile$ $\endgroup$
    – Git Gud
    Nov 26, 2018 at 9:53
  • $\begingroup$ $f'$ is the function (therefore a map), $f'(x)$ is the value of the function in the point $x$ (therefore a number). Yes, people mix them up all the times. $\endgroup$
    – gented
    Nov 26, 2018 at 11:01
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    $\begingroup$ You are correct that $f$ is the function and $f(x)$ is the value of the function when evaluated at a point $x$ in its domain (ditto $f'$ and $f'(x)$). Many elementary texts blur this distinction in an attempt to "dumb down" the material. This causes no end of confusion later on, and you have done well to note the problem. $\endgroup$
    – Xander Henderson
    Nov 26, 2018 at 13:20
  • $\begingroup$ @J.Smith In case my answer would be deleted I let here the main reference I've found on that topic Calculus for Dummies $\endgroup$
    – user
    Nov 26, 2018 at 14:36

4 Answers 4

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By definition a function is a triple $(f,D,C)$, which is very often denoted by $f \colon D \to C$, where $C,D$ are two sets and $f$ associates to each element of $D$ one and only one element of $C$.

So when it is clear what $C$ and $D$ are, or in cases where it is not possible or not necessary to write them down, you just write $f$. The expression $f(x)$ denotes the element in $C$ which $x \in D$ is mapped to. So $f$ is a function, $f(x)$ is an element of $C$, two completely different things.

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  • $\begingroup$ Nice answer! (+1) $\endgroup$
    – Robert Z
    Nov 26, 2018 at 8:52
  • $\begingroup$ Thank you @RobertZ. $\endgroup$
    – Gibbs
    Nov 26, 2018 at 8:59
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    $\begingroup$ I agree with the forst part (+1) but I would be more relaxed for the second part, I think that using f(x) to indicate the function can be tolerated. $\endgroup$
    – user
    Nov 26, 2018 at 9:22
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    $\begingroup$ I think it is a good habit for a beginner to stick to the definitions. After some practice and experience one becomes conscious on where conventions can be relaxed and some abuse of language might be ok. $\endgroup$
    – Gibbs
    Nov 26, 2018 at 10:06
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    $\begingroup$ The distinction between $f$ and $f(x) $ is important but "let $f(x) =x\sin(1/x)$" has almost become a routine shorthand for "let $f:\mathbb{R} \setminus\{0\}\to\mathbb {R} $ be a function defined by $f(x) =x\sin(1/x)$". $\endgroup$
    – Paramanand Singh
    Nov 27, 2018 at 5:05
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$f$ denotes the function and $f(x)$ the output of the function when evaluated at $x$.

This convention does not differ for the derivative.

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  • $\begingroup$ f’ denotes the derivative and f’(x) denotes the output of the derivative, which is the instantaneous rate of change of a function at any point. Correct? $\endgroup$
    – user618086
    Nov 26, 2018 at 10:08
  • $\begingroup$ @J.Smith: you get it. $\endgroup$
    – user65203
    Nov 26, 2018 at 10:13
  • $\begingroup$ @YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :) $\endgroup$
    – user
    Nov 26, 2018 at 10:17
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    $\begingroup$ @gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no. $\endgroup$
    – user65203
    Nov 26, 2018 at 10:20
  • $\begingroup$ @YvesDaoust Also f creates ambiguity if we do not specify the domain and the codomain, therefore anytime we refer to a function we should use $f:A\to B$. I don't thing it would be a useful notation. $f(x)$ to indicate the funtion can be used many times without any ambiguity. Of course we ca suggest to do not use that but it is a matter of preferences and not a law. $\endgroup$
    – user
    Nov 26, 2018 at 10:23
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The derivative of the function $f$ is $f'$. People usually make the mistake of saying that it is $f'(x)$, just like they talk about, say, the function $\sin(x)$, when, in fact, they should be talking about the $\sin$ function.

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  • $\begingroup$ So when people say 2x is the derivative of x^2, is that incorrect? Also, am I correct in saying that the derivative is the function of the form y=f’(x) whose output represents the instantaneous rate of change at any point of a function or the slope of the tangent line to a point on a curve? $\endgroup$
    – user618086
    Nov 26, 2018 at 8:38
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    $\begingroup$ It is an abuse of language. It would be correct to say that $(\operatorname{Id}^2)'=2\operatorname{Id}$ and, of course, that's what people mean when they say that the derivative of $x^2$ is $2x$. Concerning your second question, the answer is affirmative. $\endgroup$ Nov 26, 2018 at 8:42
  • $\begingroup$ This is too far away from the original question and I suggest that you post it as another question. But, for me (and, I think, for most users of this forum)$$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}.$$ $\endgroup$ Nov 26, 2018 at 8:59
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    $\begingroup$ I have to disagree with José when he says that "of course, that's what people mean when they say that the derivative of $x^2$ is $2x$". I agree that's what people mean when they know what they're talking about and understand the distinction outlined in several answers in this question. The thing is, most people that say this do not understand this distinction, at least I remember a time (pre-university) where I didn't since I didn't even know of a proper definition of function. $\endgroup$
    – Git Gud
    Nov 26, 2018 at 9:47
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    $\begingroup$ @TaylorRendon Yes, that gives a good idea about the meaning of derivative of a function at a point. $\endgroup$ Jan 1, 2021 at 23:18
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I would read $f'(x)$ as "the function $f'$ applied to the element $x$ of the domain". This gives us a a new element in the range. Meanwhile I read $f'$ as a relation, it tells us which elements are mapped to which other elements. The prime just tells us that it is relation to some other function $f$ in a very specific way (derivation).

Example: Our 'input' set is $\{1,2,3 \}$ our output set is $\{A,B,C,D\}$ $$f=\{(1,C),(2,A),(3,D) \}$$ So we now know that $f(1)=C$ and $f(2)=A$. Notice that the element $B$ is not reached and this function is not surjective.

What you should take from this finite example is that a function is a rule that tells us which elements are in a way "paired", while $f(x)$ tells us about a specific pair. However sometimes people just represent the function like this by saying:

For arbitrary $x$ (so in our example $1$,$2$ or $3$), $f(x)$ is given by $\dots$ This is indeed another representation of the same idea, but mathematicians ofter prefer the "relation" idea.

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