We are all used to seeing $y=f(x)$ where we wish to plot the function $f$ on the $xy$ plane. We can differentiate both sides, $\frac{d}{dx}(y) = \frac{d}{dx}(f(x))$ to get the good old $\frac{dy}{dx} = f'(x)$, now we know that $\frac{dy}{dx}$ and $f'(x)$ mean the same (the derivative of $f$), but also I have seen $\frac{df}{dx}$ written. First off, is this shorthand for $\frac{df(x)}{dx}$ and second of all, in the statement $y=f(x)$ we know $y$ is not a function, its just a variable that takes the value of the function $f$ at each point $x$. That means we can translate $\frac{dy}{dx}$ to mean "the rate of change of the variable $y$ as $x$ varies". And we can translate $\frac{df(x)}{dx}$ to mean the same, as $y$ and $f(x)$ mean the exact same thing, they are both ouputs of the function $f$ (and we are interested how fast/slow these outputs change as we vary the input $x$). Taking a look at $\frac{df}{dx}$ one might interpret it to mean, "the rate of change of the variable $f$ as $x$ varies" but that's wrong, its actually "the rate of change of the output of $f$ as $x$ varies). Why do mathematicans not just write $\frac{df(x)}{dx}$ so there is no chance someone interprets $f$ as a variable?
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2I’m voting to close this question because we don't need yet again an endless discussion about a topic like this. – Kurt G. Aug 15 '23 at 19:22
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4Does this answer your question? Notation for the derivative of a function: $f'$ or $f'(x);$? – Jam Aug 15 '23 at 19:26
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2The answer you accepted is actually wrong, as $\frac{dy}{dx}$ cannot possibly be a function, under any reasonable interpretation of $x$. Just ask yourself, what is $x$ there? It does not appear in $f'$, which is a pure function. Everything you wrote in your question is correct, so I'm surprised that you accepted a problematic answer.. – user21820 Aug 16 '23 at 07:15
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I like the first paragraph only of the answer upon further inspection! $\frac{df(x)}{dx} > \frac{df}{dx}$. Don't take this inequality too seriously. – Nav Bhatthal Aug 16 '23 at 07:18
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The reason for the common abuse of notation is partly historical; the notion of a function in the (standard) rigorous sense is a very recent one. Earlier, variables were really used to represent quantities that vary, and some people decided that it was convenient/useful to tell others what it varied with, so they might write $f = f(x,y,z)$, which is obviously not rigorous if you wish to treat $f$ as referring to a specific (unambiguous) entity/concept. But this abuse of notation has stuck around, unfortunately... – user21820 Aug 16 '23 at 07:19
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I don't think enough mathematicians care for it to be a problem that's taken seriously – Nav Bhatthal Aug 16 '23 at 07:21
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Yes, they often wouldn't, because (usually) they do know precisely what they are thinking in their head. And other mathematicians can (usually) figure out what it is too. It's only students who are the recipients of the confusion. However, some do care, so it's kind of up to you to choose how seriously to take it. I personally don't use bad notation, but that's me. – user21820 Aug 16 '23 at 07:25
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Part of the issue stems from the fact that rigour and proof writing is barely barely assessed at school level so most mathematicians leave school not having a good formal introduction to it. – Nav Bhatthal Aug 16 '23 at 11:03
2 Answers
I agree with you. But, this abuse of notation (conflating a function $f$ with its value $f(x)$) appears all over mathematics, and usually it's innocent enough as there's no ambiguity. But you are correct to note the distinction.
If we're being careful, the derivative of a function is a function, so we probably ought to write something like $$ f' = \frac{dy}{dx} $$ and if we choose to evaluate at a particular $x=a$, say, $$ f'(a) = \frac{dy}{dx}\bigg|_{x=a}. $$ This is annoying because it requires an additional symbol when we intend for the $x$ in $f(x)$ to be generic anyway.
At the end of the day, context is everything. Notation works for you, not the other way around. When you write an expression, ask yourself is this a function or a value of the function (or something else).
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Yeah its a very very minute distinction, but as a high-school student, I see stuff like this pop up everywhere. Proud to have spot this though! – Nav Bhatthal Aug 15 '23 at 17:45
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2"and usually it's innocent enough as there's no ambiguity" << I completely, utterly disagree with this. It is innocent enough and unambiguous... Only when used by someone who already perfectly understands derivatives, to communicate with someone who already perfectly understands derivatives. But in many cases it's used by teachers to communicate with a classful of students, which results in many students never getting more than a very vague understanding of the notation and the topic. – Stef Aug 16 '23 at 08:58
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Hey ho that's why stack exchange exists! Doing the schools work for them! Even though that's not the intent of stack exchange... – Nav Bhatthal Aug 16 '23 at 11:04
Well, because $f$ is a variable, even when it’s the name of a function. It varies. It is simply a difference in perception and appearance, not a genuine mathematical difference.
You might also see $y(x)$. And when you really need to specify when a variable is dependent or independent, it’s important to be careful with all this. But this single variable distinction you are talking about does not really matter.
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1$f$ is just a letter we use to represent a function, not its output, which we call $f(x)$ so if we want to see the output change, is it not clearer to write $\frac{df(x)}{dx}$? The function is not changing, its the value $f(x)$. – Nav Bhatthal Aug 15 '23 at 17:43
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1@Mathguy . What we mean by any of the versions of $f'$ that are discussed in this thread is clear from context since the days of Newton and Leibniz. You have a choice: rely on what people know historically when they read that notation or (if you believe they still need education): use a clumsier notation. – Kurt G. Aug 15 '23 at 19:21