Notation for the derivative of a function: $f'$ or $f'(x);$?
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?
calculus derivatives notation
edited Nov 26 at 12:12
amWhy
191k28224439
asked Nov 26 at 8:25
J. Smith
241
add a comment |
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?
calculus derivatives notation
edited Nov 26 at 12:12
amWhy
191k28224439
asked Nov 26 at 8:25
J. Smith
241
By the way, check the first bullet point here $ddot smile$
– Git Gud
Nov 26 at 9:53
$f'$ 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 at 11:01
1
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 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
– gimusi
Nov 26 at 14:36
add a comment |
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?
calculus derivatives notation
edited Nov 26 at 12:12
amWhy
191k28224439
asked Nov 26 at 8:25
J. Smith
241
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?
calculus derivatives notation
calculus derivatives notation
edited Nov 26 at 12:12
amWhy
191k28224439
asked Nov 26 at 8:25
J. Smith
241
edited Nov 26 at 12:12
amWhy
191k28224439
asked Nov 26 at 8:25
J. Smith
241
edited Nov 26 at 12:12
amWhy
191k28224439
edited Nov 26 at 12:12
amWhy
191k28224439
edited Nov 26 at 12:12
amWhy
191k28224439
191k28224439
asked Nov 26 at 8:25
J. Smith
241
asked Nov 26 at 8:25
J. Smith
241
asked Nov 26 at 8:25
J. Smith
241
241
By the way, check the first bullet point here $ddot smile$
– Git Gud
Nov 26 at 9:53
$f'$ 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 at 11:01
1
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 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
– gimusi
Nov 26 at 14:36
add a comment |
By the way, check the first bullet point here $ddot smile$
– Git Gud
Nov 26 at 9:53
$f'$ 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 at 11:01
1
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 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
– gimusi
Nov 26 at 14:36
By the way, check the first bullet point here $ddot smile$
– Git Gud
Nov 26 at 9:53
By the way, check the first bullet point here $ddot smile$
– Git Gud
Nov 26 at 9:53
$f'$ 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 at 11:01
$f'$ 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 at 11:01
1
1
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 at 13:20
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 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
– gimusi
Nov 26 at 14:36
@J.Smith In case my answer would be deleted I let here the main reference I've found on that topic Calculus for Dummies
– gimusi
Nov 26 at 14:36
add a comment |
4 Answers
4
active
oldest
votes
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.
answered Nov 26 at 8:39
Gibbs
4,7583726
Nice answer! (+1)
– Robert Z
Nov 26 at 8:52
Thank you @RobertZ.
– Gibbs
Nov 26 at 8:59
2
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.
– gimusi
Nov 26 at 9:22
1
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 at 10:06
The distinction between $f$ and $f(x) $ is important but "let $f(x) =xsin(1/x)$" has almost become a routine shorthand for "let $f:mathbb{R} setminus{0}tomathbb {R} $ be a function defined by $f(x) =xsin(1/x)$".
– Paramanand Singh
Nov 27 at 5:05
add a comment |
$f$ denotes the function and $f(x)$ the output of the function when evaluated at $x$.
This convention does not differ for the derivative.
answered Nov 26 at 10:07
Yves Daoust
124k671221
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?
– J. Smith
Nov 26 at 10:08
@J.Smith: you get it.
– Yves Daoust
Nov 26 at 10:13
@YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :)
– gimusi
Nov 26 at 10:17
2
@gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no.
– Yves Daoust
Nov 26 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 $f:Ato B$. 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.
– gimusi
Nov 26 at 10:23
|
show 9 more comments
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.
edited Nov 26 at 20:51
amWhy
191k28224439
answered Nov 26 at 8:49
Wesley Strik
1,486422
add a comment |
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.
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
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?
– J. Smith
Nov 26 at 8:38
1
It is an abuse of language. It would be correct to say that $(operatorname{Id}^2)'=2operatorname{Id}$ 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 at 8:42
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)$$f'(x)=lim_{yto x}frac{f(y)-f(x)}{y-x}.$$
– José Carlos Santos
Nov 26 at 8:59
1
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 at 9:47
@GitGud Well, okay, perhaps that I was more than a bit optimistic here…
– José Carlos Santos
Nov 26 at 9:48
|
show 4 more comments
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4 Answers
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active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
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active
oldest
votes
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.
answered Nov 26 at 8:39
Gibbs
4,7583726
Nice answer! (+1)
– Robert Z
Nov 26 at 8:52
Thank you @RobertZ.
– Gibbs
Nov 26 at 8:59
2
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.
– gimusi
Nov 26 at 9:22
1
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 at 10:06
The distinction between $f$ and $f(x) $ is important but "let $f(x) =xsin(1/x)$" has almost become a routine shorthand for "let $f:mathbb{R} setminus{0}tomathbb {R} $ be a function defined by $f(x) =xsin(1/x)$".
– Paramanand Singh
Nov 27 at 5:05
add a comment |
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.
answered Nov 26 at 8:39
Gibbs
4,7583726
Nice answer! (+1)
– Robert Z
Nov 26 at 8:52
Thank you @RobertZ.
– Gibbs
Nov 26 at 8:59
2
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.
– gimusi
Nov 26 at 9:22
1
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 at 10:06
The distinction between $f$ and $f(x) $ is important but "let $f(x) =xsin(1/x)$" has almost become a routine shorthand for "let $f:mathbb{R} setminus{0}tomathbb {R} $ be a function defined by $f(x) =xsin(1/x)$".
– Paramanand Singh
Nov 27 at 5:05
add a comment |
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.
answered Nov 26 at 8:39
Gibbs
4,7583726
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.
answered Nov 26 at 8:39
Gibbs
4,7583726
edited Nov 26 at 9:00
answered Nov 26 at 8:39
Gibbs
4,7583726
answered Nov 26 at 8:39
Gibbs
4,7583726
answered Nov 26 at 8:39
Gibbs
4,7583726
4,7583726
Nice answer! (+1)
– Robert Z
Nov 26 at 8:52
Thank you @RobertZ.
– Gibbs
Nov 26 at 8:59
2
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.
– gimusi
Nov 26 at 9:22
1
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 at 10:06
The distinction between $f$ and $f(x) $ is important but "let $f(x) =xsin(1/x)$" has almost become a routine shorthand for "let $f:mathbb{R} setminus{0}tomathbb {R} $ be a function defined by $f(x) =xsin(1/x)$".
– Paramanand Singh
Nov 27 at 5:05
add a comment |
Nice answer! (+1)
– Robert Z
Nov 26 at 8:52
Thank you @RobertZ.
– Gibbs
Nov 26 at 8:59
2
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.
– gimusi
Nov 26 at 9:22
1
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 at 10:06
The distinction between $f$ and $f(x) $ is important but "let $f(x) =xsin(1/x)$" has almost become a routine shorthand for "let $f:mathbb{R} setminus{0}tomathbb {R} $ be a function defined by $f(x) =xsin(1/x)$".
– Paramanand Singh
Nov 27 at 5:05
Nice answer! (+1)
– Robert Z
Nov 26 at 8:52
Nice answer! (+1)
– Robert Z
Nov 26 at 8:52
Thank you @RobertZ.
– Gibbs
Nov 26 at 8:59
Thank you @RobertZ.
– Gibbs
Nov 26 at 8:59
2
2
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.
– gimusi
Nov 26 at 9:22
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.
– gimusi
Nov 26 at 9:22
1
1
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 at 10:06
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 at 10:06
The distinction between $f$ and $f(x) $ is important but "let $f(x) =xsin(1/x)$" has almost become a routine shorthand for "let $f:mathbb{R} setminus{0}tomathbb {R} $ be a function defined by $f(x) =xsin(1/x)$".
– Paramanand Singh
Nov 27 at 5:05
The distinction between $f$ and $f(x) $ is important but "let $f(x) =xsin(1/x)$" has almost become a routine shorthand for "let $f:mathbb{R} setminus{0}tomathbb {R} $ be a function defined by $f(x) =xsin(1/x)$".
– Paramanand Singh
Nov 27 at 5:05
add a comment |
$f$ denotes the function and $f(x)$ the output of the function when evaluated at $x$.
This convention does not differ for the derivative.
answered Nov 26 at 10:07
Yves Daoust
124k671221
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?
– J. Smith
Nov 26 at 10:08
@J.Smith: you get it.
– Yves Daoust
Nov 26 at 10:13
@YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :)
– gimusi
Nov 26 at 10:17
2
@gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no.
– Yves Daoust
Nov 26 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 $f:Ato B$. 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.
– gimusi
Nov 26 at 10:23
|
show 9 more comments
$f$ denotes the function and $f(x)$ the output of the function when evaluated at $x$.
This convention does not differ for the derivative.
answered Nov 26 at 10:07
Yves Daoust
124k671221
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?
– J. Smith
Nov 26 at 10:08
@J.Smith: you get it.
– Yves Daoust
Nov 26 at 10:13
@YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :)
– gimusi
Nov 26 at 10:17
2
@gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no.
– Yves Daoust
Nov 26 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 $f:Ato B$. 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.
– gimusi
Nov 26 at 10:23
|
show 9 more comments
$f$ denotes the function and $f(x)$ the output of the function when evaluated at $x$.
This convention does not differ for the derivative.
answered Nov 26 at 10:07
Yves Daoust
124k671221
$f$ denotes the function and $f(x)$ the output of the function when evaluated at $x$.
This convention does not differ for the derivative.
answered Nov 26 at 10:07
Yves Daoust
124k671221
answered Nov 26 at 10:07
Yves Daoust
124k671221
answered Nov 26 at 10:07
Yves Daoust
124k671221
answered Nov 26 at 10:07
Yves Daoust
124k671221
124k671221
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?
– J. Smith
Nov 26 at 10:08
@J.Smith: you get it.
– Yves Daoust
Nov 26 at 10:13
@YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :)
– gimusi
Nov 26 at 10:17
2
@gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no.
– Yves Daoust
Nov 26 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 $f:Ato B$. 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.
– gimusi
Nov 26 at 10:23
|
show 9 more comments
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?
– J. Smith
Nov 26 at 10:08
@J.Smith: you get it.
– Yves Daoust
Nov 26 at 10:13
@YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :)
– gimusi
Nov 26 at 10:17
2
@gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no.
– Yves Daoust
Nov 26 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 $f:Ato B$. 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.
– gimusi
Nov 26 at 10:23
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?
– J. Smith
Nov 26 at 10:08
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?
– J. Smith
Nov 26 at 10:08
@J.Smith: you get it.
– Yves Daoust
Nov 26 at 10:13
@J.Smith: you get it.
– Yves Daoust
Nov 26 at 10:13
@YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :)
– gimusi
Nov 26 at 10:17
@YvesDaoust I think we could be more "relaxed" with that definition and notation, no one will be wound considering the function $f(x)$ :)
– gimusi
Nov 26 at 10:17
2
2
@gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no.
– Yves Daoust
Nov 26 at 10:20
@gimusi: this would reduce the expressive power (not possible to distinguish the function and the value) and create ambiguities. So, no.
– Yves Daoust
Nov 26 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 $f:Ato B$. 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.
– gimusi
Nov 26 at 10:23
@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 $f:Ato B$. 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.
– gimusi
Nov 26 at 10:23
|
show 9 more comments
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.
edited Nov 26 at 20:51
amWhy
191k28224439
answered Nov 26 at 8:49
Wesley Strik
1,486422
add a comment |
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.
edited Nov 26 at 20:51
amWhy
191k28224439
answered Nov 26 at 8:49
Wesley Strik
1,486422
add a comment |
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.
edited Nov 26 at 20:51
amWhy
191k28224439
answered Nov 26 at 8:49
Wesley Strik
1,486422
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.
edited Nov 26 at 20:51
amWhy
191k28224439
answered Nov 26 at 8:49
Wesley Strik
1,486422
edited Nov 26 at 20:51
amWhy
191k28224439
edited Nov 26 at 20:51
amWhy
191k28224439
edited Nov 26 at 20:51
amWhy
191k28224439
191k28224439
answered Nov 26 at 8:49
Wesley Strik
1,486422
answered Nov 26 at 8:49
Wesley Strik
1,486422
answered Nov 26 at 8:49
Wesley Strik
1,486422
1,486422
add a comment |
add a comment |
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.
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
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?
– J. Smith
Nov 26 at 8:38
1
It is an abuse of language. It would be correct to say that $(operatorname{Id}^2)'=2operatorname{Id}$ 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 at 8:42
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)$$f'(x)=lim_{yto x}frac{f(y)-f(x)}{y-x}.$$
– José Carlos Santos
Nov 26 at 8:59
1
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 at 9:47
@GitGud Well, okay, perhaps that I was more than a bit optimistic here…
– José Carlos Santos
Nov 26 at 9:48
|
show 4 more comments
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.
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
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?
– J. Smith
Nov 26 at 8:38
1
It is an abuse of language. It would be correct to say that $(operatorname{Id}^2)'=2operatorname{Id}$ 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 at 8:42
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)$$f'(x)=lim_{yto x}frac{f(y)-f(x)}{y-x}.$$
– José Carlos Santos
Nov 26 at 8:59
1
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 at 9:47
@GitGud Well, okay, perhaps that I was more than a bit optimistic here…
– José Carlos Santos
Nov 26 at 9:48
|
show 4 more comments
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.
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
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.
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
answered Nov 26 at 8:27
José Carlos Santos
149k22117219
149k22117219
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?
– J. Smith
Nov 26 at 8:38
1
It is an abuse of language. It would be correct to say that $(operatorname{Id}^2)'=2operatorname{Id}$ 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 at 8:42
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)$$f'(x)=lim_{yto x}frac{f(y)-f(x)}{y-x}.$$
– José Carlos Santos
Nov 26 at 8:59
1
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 at 9:47
@GitGud Well, okay, perhaps that I was more than a bit optimistic here…
– José Carlos Santos
Nov 26 at 9:48
|
show 4 more comments
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?
– J. Smith
Nov 26 at 8:38
1
It is an abuse of language. It would be correct to say that $(operatorname{Id}^2)'=2operatorname{Id}$ 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 at 8:42
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)$$f'(x)=lim_{yto x}frac{f(y)-f(x)}{y-x}.$$
– José Carlos Santos
Nov 26 at 8:59
1
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 at 9:47
@GitGud Well, okay, perhaps that I was more than a bit optimistic here…
– José Carlos Santos
Nov 26 at 9:48
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?
– J. Smith
Nov 26 at 8:38
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?
– J. Smith
Nov 26 at 8:38
1
1
It is an abuse of language. It would be correct to say that $(operatorname{Id}^2)'=2operatorname{Id}$ 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 at 8:42
It is an abuse of language. It would be correct to say that $(operatorname{Id}^2)'=2operatorname{Id}$ 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 at 8:42
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)$$f'(x)=lim_{yto x}frac{f(y)-f(x)}{y-x}.$$
– José Carlos Santos
Nov 26 at 8:59
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)$$f'(x)=lim_{yto x}frac{f(y)-f(x)}{y-x}.$$
– José Carlos Santos
Nov 26 at 8:59
1
1
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 at 9:47
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 at 9:47
@GitGud Well, okay, perhaps that I was more than a bit optimistic here…
– José Carlos Santos
Nov 26 at 9:48
@GitGud Well, okay, perhaps that I was more than a bit optimistic here…
– José Carlos Santos
Nov 26 at 9:48
|
show 4 more comments
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By the way, check the first bullet point here $ddot smile$
– Git Gud
Nov 26 at 9:53
$f'$ 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 at 11:01
1
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 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
– gimusi
Nov 26 at 14:36