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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 '18 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 '18 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 '18 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 $\endgroup$ – gimusi Nov 26 '18 at 14:36
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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 '18 at 8:52
  • $\begingroup$ Thank you @RobertZ. $\endgroup$ – Gibbs Nov 26 '18 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. $\endgroup$ – gimusi Nov 26 '18 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 '18 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 '18 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 '18 at 10:08
  • $\begingroup$ @J.Smith: you get it. $\endgroup$ – Yves Daoust Nov 26 '18 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 $f(x)$ :) $\endgroup$ – gimusi Nov 26 '18 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. $\endgroup$ – Yves Daoust Nov 26 '18 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. $\endgroup$ – gimusi Nov 26 '18 at 10:23
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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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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 '18 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. $\endgroup$ – José Carlos Santos Nov 26 '18 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)$$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}.$$ $\endgroup$ – José Carlos Santos Nov 26 '18 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 '18 at 9:47
  • $\begingroup$ @GitGud Well, okay, perhaps that I was more than a bit optimistic here… $\endgroup$ – José Carlos Santos Nov 26 '18 at 9:48

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