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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?

amWhy
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  • By the way, check the first bullet point here $\ddot \smile$ – Git Gud Nov 26 '18 at 09:53
  • $f'$ 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. – gented Nov 26 '18 at 11:01
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    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. – Xander Henderson Nov 26 '18 at 13:20
  • @J.Smith In case my answer would be deleted I let here the main reference I've found on that topic Calculus for Dummies – user Nov 26 '18 at 14:36

4 Answers4

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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.

Gibbs
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  • Nice answer! (+1) – Robert Z Nov 26 '18 at 08:52
  • Thank you @RobertZ. – Gibbs Nov 26 '18 at 08:59
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    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. – user Nov 26 '18 at 09:22
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    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. – Gibbs Nov 26 '18 at 10:06
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    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 $f(x) =x\sin(1/x)$". – Paramanand Singh Nov 27 '18 at 05:05
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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.

  • 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? –  Nov 26 '18 at 10:08
  • @J.Smith: you get it. –  Nov 26 '18 at 10:13
  • @YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :) – user Nov 26 '18 at 10:17
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    @gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no. –  Nov 26 '18 at 10:20
  • @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. – user Nov 26 '18 at 10:23
  • @YvesDaoust I've just given a authorative reference about that! – user Nov 26 '18 at 10:24
  • @YvesDaoust That we specify the $x$ variable for example and sometimes it can be useful. – user Nov 26 '18 at 10:28
  • @YvesDaoust Let consider $f(x)=\cos(x^2+1)$, the what is the ambiguity if we state that the derivative function of the function $f(x)$ is the function $f'(x)=-2x\sin(x^2+1)$? I really can't see any problem here. – user Nov 26 '18 at 10:36
  • @gimusi If you want to talk about "the derivative of the function $f(x)$", then you need to concede that it can be denoted as $(f(x))'$. But you can't have $(f(x))'$ and $f'(x)$ mean the same thing. This is just one of many problems. – Git Gud Nov 26 '18 at 10:45
  • @GitGud I'm not claiming that your notation or suggestions are wrong but they are indeed notations and conventions and you should also admit the existence of other different notations/conventions. Once the convention/notatin is clear I don't see any problem to use that. – user Nov 26 '18 at 10:48
  • @gimusi: please stop that harassment. –  Nov 26 '18 at 10:50
  • @gimusi You asked Yves a question, I gave you an answer. Regardless of what you just said, this is an answer to your question. – Git Gud Nov 26 '18 at 10:50
  • @YvesDaoust Sorry that wasn't my intention! I like those kind of open discussion to clarify different point of views. I can delete that if it is a problem. – user Nov 26 '18 at 10:52
  • @GitGud Ok, thanks for the clarification. Bye – user Nov 26 '18 at 10:52
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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.

José Carlos Santos
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  • 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? –  Nov 26 '18 at 08:38
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    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. – José Carlos Santos Nov 26 '18 at 08:42
  • 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)$$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}.$$ – José Carlos Santos Nov 26 '18 at 08:59
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    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. – Git Gud Nov 26 '18 at 09:47
  • @GitGud Well, okay, perhaps that I was more than a bit optimistic here… – José Carlos Santos Nov 26 '18 at 09:48
  • Hi GitGud, would it be more appropriate to say that the derivative of the function “f” is the function y=f’(x) as opposed to just “f’(x)” –  Nov 26 '18 at 09:54
  • @J.Smith No, $y=f'(x)$ isn't a function, it's an equality. If you want to be strict and protect yourself from pedantic people like myself, you can say that the derivative of $f$ is $f'$, that's it. – Git Gud Nov 26 '18 at 10:04
  • Ok so, my textbook often says “the function y=f(x)” are they really trying to say “the function f, represented by the rule “y=f(x)”? –  Nov 26 '18 at 10:07
  • I think it depends on the context (that's one of the downsides of using wrong notation). It's possible the author is thinking about graphs that he means to say that while $x$ will be represented on the horizontal axis, $f(x)$ will be on the vertical axis (or in the $y$-axis), hence $y=f(x)$, I will stop answering your questions in José's answer, because every time we comment here, he is pinged. – Git Gud Nov 26 '18 at 10:34
  • @JoséCarlosSantos - I have a quick question after reading through the comments; would it be correct to say: “the derivative of a function measures the rate of change of the function value with respect to a change in its argument”? – Taylor Rendon Jan 01 '21 at 23:15
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    @TaylorRendon Yes, that gives a good idea about the meaning of derivative of a function at a point. – José Carlos Santos Jan 01 '21 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.

amWhy
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