I'm currently teaching a couple of courses that have a calculus prerequisite, and within the last week I've had two students make notational mistakes that amount to writing $y\frac{d}{dx}$ rather than $\frac{d}{dx}y$ (although in terms of different variables than $y$ and $x$). E.g., they might write something like

$$ x^2 \frac{d}{dx}=2x.$$

At least one of them actually seemed relatively fluent with calculus, in the sense of knowing facts like $\int x^n dx=x^{n+1}/(n+1)$. I pointed out his mistake and made the analogy with the ungrammatical expression $4\sqrt{}$. I mentioned that I had just had another student make a similar mistake recently, and I asked him if he could explain more about what he was thinking when he wrote it. His response was that that was just the way he had gotten used to seeing it in his calculus book! (I assume this is not actually true.)

Can anyone provide any insight into why this would be a common enough mistake that I would see it this frequently? I wonder if there is some confusion because the $f'$ notation involves a postfix operator, or because in an integral we usually write the $dx$ at the end...?

It would seem obvious to me that their notation wouldn't make sense if you think of the derivative as the ratio of two infinitesimals, but presumably many of them haven't been exposed to that way of thinking. Part of their confusion may also be because they are used to everything being $y(x)$, never $\Psi(x)$ or $v(t)$. They also never seem to have been asked to think about the meaning of notation and why it makes sense, and don't seem to understand attempts to elicit discussion of this kind of thing. E.g., if I ask them whether they've been taught that $d/dx$ is an operator, like $\sqrt{}$ or $\sin{}$, they generally look at me blankly, as though this is not the kind of thing that was ever discussed in their calculus class.

  • $\begingroup$ Can you give the context? Exact differentials? Also is yd/dx (second form you list) the same as dy/dx. $\endgroup$ – guest Dec 3 '18 at 22:52
  • $\begingroup$ @guest: The context in both cases is just that they're given two variables and they need to notate the derivative, like $d\Psi/dx$ or $dv/dt$. Nothing fancier than that. Also is yd/dx (second form you list) the same as dy/dx. There is nothing appearing to the right of the $yd/dx$, so it's just gobbledegook, it doesn't mean anything. $\endgroup$ – Ben Crowell Dec 3 '18 at 22:57
  • $\begingroup$ Explain to them that it is simply a notation, not a product. Have you tried using a different notation, like prime? I find notation $\frac{d}{dx}y$ a bit unwieldy, especially if you have only one independent variable, and the relationship is obvious. $$ (x^2)' =2x.$$ is so much simpler. $\endgroup$ – Rusty Core Dec 4 '18 at 0:38
  • $\begingroup$ I don't see why you would use either. Just use dy/dx. If you need to show d/dx as an operator, do so with a parens. Like d/dx(45x) or the like. But I would avoid the way you wrote the second one, if the only argument is y, and with no parens.. I'm sure you will reply that it is right, but my point is not that...just confusing. $\endgroup$ – guest Dec 4 '18 at 15:15
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    $\begingroup$ As a shot-in-the-dark guess, I wonder if this relates to years and years experience with lower level math in which $2 + 3$ might be read as something like "take $2$ then add to it ..." and $4 \times 3$ might be read as something like "take 4 and then multiply by ...", whereas situations in which the operation is to the left (like trig. functions and logarithms) are still very new things that they have very little experience with? Yes, I realize WE might think of $2 +$ as the operation, but maybe not beginning students. $\endgroup$ – Dave L Renfro Dec 5 '18 at 22:25

When a student writes incorrect notation, ask them to read it out loud. I would say something like:

Something here doesn't look right, but we can fix it. Could you read this work out loud? I think probably you are not super familiar with this topic, and that's okay, but this can help us fix it.

I had success with this when dealing with a student who wrote "$\sqrt 4 = \sqrt 2$". After some prodding and reassurance, they read it to me: the student said "The question is the square root of 4. The answer is the square root is 2." The key here was that the student didn't understand what the "$=$" meant (which is a whole topic of discussion on its own) nor did the student really understand what the "$\sqrt{}$" meant (see also this question which is the source of my understanding).

But in this case you may be surprised by something similar. Maybe the student will say that "$x^2 \frac{d}{dx} = 2x$" says "$x^2$ prime is $2x$"? Or slightly better "$x^2$'s derivative is $2x$". If the student reads "d d x," don't accept that: get them to translate what it means to them inside the sentence. Maybe they think "$\frac{d}{dx}=$" is a group of symbols that separates a question and an answer, like my student above.

No matter what they say, the correction is relatively easy: you can explain that $\frac{d}{dx}$ actually precisely means "The derivative of." Then you can read their work back to them for effect to help them see why it looks strange: "$x^2$ the derivative of equals $2x$." It's important that you emphasize this is not your opinion, but that these symbols have an accepted precise meaning.

I hope this helps.

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    $\begingroup$ Thanks, these are some interesting thoughts about how to draw students out, which is difficult to do. But I think there are some apples-oranges comparisons here. The student mistake in writing $\sqrt{4}=\sqrt{2}$ is a logical/semantic mistake; the assertion the student writes has a well-defined truth value, which happens to be that it's false. The mistake in $x^2\frac{d}{dx}=2x$ is a syntactical mistake. It has no well-defined truth value, but this is because of an essentially arbitrary syntactical convention, which is that $d/dx$ is a prefix rather than a postfix operator.[...] $\endgroup$ – Ben Crowell Dec 5 '18 at 2:12
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    $\begingroup$ [...] I'm not sure that translating back and forth to English is something we should encourage, since the syntax of English is not isomorphic to the syntax of mathematical expressions. Even if we could find a translation convention that worked, the student would then have to memorize one translation convention for the prefix $d/dx$ operator, and another one for the postfix "prime" notation. $\endgroup$ – Ben Crowell Dec 5 '18 at 2:14
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    $\begingroup$ @BenCrowell While correct, your distinction seems not very relevant: with the explanation, we see that $\sqrt{4}=\sqrt{2}$ was syntaxically well-formed by pure accident, the use of each symbol not being understood. Here, I feel that a reasoning akin to the one seen when using a calculator (write the expression, then type on the differentiation operator) could explain what you observed. $\endgroup$ – Benoît Kloeckner Dec 5 '18 at 11:19
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    $\begingroup$ I agree with Ben Crowell, it's an arbitrary convention to write certain operators postfix and others prefix, and it cannot be decided by reading things out. Suppose a student said "$x^2\frac{d}{dx}$ is to be read as $x^2$ derived with respect to $x$"? Or imagine a student concludes "So when I say the square of $x$, I should write $^2x$, while when I say $x$ squared I have to write $x^2$?". $\endgroup$ – Michael Bächtold Dec 6 '18 at 22:01
  • $\begingroup$ @Michael Bächtold: Great comment! (your most recent) I've actually had students say things somewhat like this in other contexts (square of $x$ vs. $x$ squared), namely a perspective that I had never considered before that also makes perfectly good sense. This will be a really good student who is genuinely confused, or a really good student who is being mischievous. Sometimes early during a semester, a student who I had not thought was especially thoughtful, will say something like this, and then my opinion changes and I also retroactively notice things that I'd overlooked about the student. $\endgroup$ – Dave L Renfro Dec 7 '18 at 12:19

I see the same mistake in my Calculus students' work a lot. And from my observations, I think there's a very simple reason for that — for typical students (here in the U.S.) this is multiplication, and in multiplication order doesn't matter. In fact, for typical students (…) almost everything is multiplication, including function notation; and they simply extend their writing habits to this new symbol without giving it much thought. And I'm going to demonstrate below that I mentioned function notation for a reason.

Disclaimer: described below are my personal observations and opinions; I can't provide any supporting literature. But then, I have years' worth of these observations, which are consistent across hundreds of students I've taught and tutored.

Imagine a middle school student needs to solve an equation like $$\frac{x}{5}=3.$$ Of course, the approach is to multiply both sides by $5$. And the typical way students are going to show their work on paper is by writing the fact that they are multiplying by $5$ by adding some markings in the same line, and then the result in the next line: $$\begin{align} \color{blue}{5\times}\frac{x}{5}&=3\color{blue}{\times5} \\ x&=15 \end{align}$$

As part of the reason for students developing this habit, I also blame the tradition (habit? expectation?) of showing what they are going to do in the same line. There's no room between the equals sign and the $3$ to fit "$\color{blue}{5\times}$" in there, so writing this multiplication from the left on the left and from the right on the right is basically the only way to fit it. And at this point it's fine. Even though pedagogically it would be much more preferable to indicate multiplication from the same side on both sides of the equation, in the spirit of doing the same thing to both sides of an equation, arithmetically this works and leads to the correct answer. (Of course, then we really have to deal with this problem in our abstract algebra classes when solving or manipulating equations in non-abelian groups.)

But then we move on to more advanced classes and encounter equations such as $$e^x=10.$$ Don't forget that for most U.S. students (deep sigh…) $f(x)$ means "$f$ times $x$", $\sin(x)$ means "sine times $x$", etc. Yeah, function notation is another sore topic… So, by following the same recipe — and because there's no room between the equals sign and the $10$ anyway! — students "multiply" both sides by $\ln$: $$\begin{align} \color{blue}{\ln}e^x&=10\color{blue}{\ln} \\ x&=\ln10 \end{align}$$

And then this pattern continues. One of the first instances when students have to write $\dfrac{d}{dx}$ themselves (as opposed to a given problem with this notation) is in the topic of implicit differentiation. It's the same situation again: given an equation, perform some steps to get an answer, and so that's what they do. For example, starting with something like $$x^2+y^2=10$$ they "multiply" both sides by $\dfrac{d}{dx}$: $$\begin{align} \color{blue}{\frac{d}{dx}}(x^2+y^2)&=10\color{blue}{\frac{d}{dx}} \\ 2x+\cdots&=\cdots \end{align}$$ (where lots of different things may appear in place of those dots), and then continue to the subsequent steps.

Of course, as others have already said, a deeper reason is not understanding the differential operator $\dfrac{d}{dx}$ as an operator, not treating mathematical notation properly in general, etc. But I wanted to address the specific question of where this particular mis-notation comes from, in my opinion.

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    $\begingroup$ @RustyCore: No, not exactly. From that Feynman's quote it seems pretty clear that he knew what $f(x)$ means (in the traditional notation), but also recognized that it may look ambiguous and confusing, and thus didn't like it. That's very different from the majority of our current students (in the U.S.) who actually don't know and don't understand (the traditional) function notation. $\endgroup$ – zipirovich Dec 22 '18 at 3:15
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    $\begingroup$ I think you're right about the immediate source of the problem, but not the true underlying source, which is a typing error of the most unacceptable kind. People do not think that "radius of circle" means "radius times circle". So why should they think that "$f(x)$" means "$f × x$"? Because they were never taught to always identify the type of every object, and never taught precisely what operations can be performed on what types of objects. Since $\ln$ is not a real number, it would obviously be meaningless to multiply by $\ln$. And "$10\ln$" is ungrammatical just like "circle radius of". $\endgroup$ – user21820 Jun 10 '19 at 17:39
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    $\begingroup$ @user21820 $10\ln$ is fine to me. It is the function that, when applied to $x \in \mathbb{R}$ returns $10\ln(x)$. $\endgroup$ – Steven Gubkin Oct 3 '19 at 19:28
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    $\begingroup$ @StevenGubkin: Haha, you and I know that we can define the action of reals on real-valued functions via pointwise scaling. However, before defining such stuff, we had better make sure that students understand more basic stuff such as basic type-checking. $\endgroup$ – user21820 Oct 4 '19 at 3:43
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    $\begingroup$ +1 for observing the damage that sloppy equation-solving writing practice gets us in. Re: @user21820's comments on grammar and types, which I fully agree with -- note that basic grammar is no longer taught in U.S. grammar/middle school, so trying to have that (critical) conversation is likely to generate more blank stares. My current college students usually can't identify a noun or verb in a sentence. (And: implicitly typed programming languages are pretty common now, too, e.g., javascript, python, etc.) $\endgroup$ – Daniel R. Collins Dec 25 '19 at 19:28

I didn't like f(x) — that looked to me like f times x. I also didn't like $\frac{dy}{dx}$ — you have a tendency to cancel the d's. So I made a different sign, something like an & sign. For logarithms it was a big L extended to the right, with the thing you take the log of inside, and so on.

I thought my symbols were just as good, if not better, than the regular symbols — it doesn't make any difference what symbols you use — but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, "What the hell are those?" I realized then that if I'm going to talk to anybody else, I'll have to use the standard symbols, so I eventually gave up my own symbols.

~ Richard Feynman

So, explain to your students that this is an accepted notation for differentiation, nothing more. Explain, that it should be taken as a single operator, not a fraction; that d's cannot be canceled. Show them other notations and allow using them even if your course and the textbooks stick to Leibniz's notation.

I think that Lagrange's notation is much cleaner for single-variable derivative than Leibniz's:

$$ (x^2)'=2x$$

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    $\begingroup$ I never understood why people use Lagrange notation at all. You have to ditch it when you get to integration anyways. And in physics where the nonrigorous manipulations and cancellations take place it's even more enjoyable. You know, $v=\frac{dx}{dt}$ and $\int v dt = \int \frac{dx}{dt}dt = \int dx = x$. Such a beatiful and simple symmetry that Lagrange notation doesn't reveal at all. $\endgroup$ – Džuris Dec 27 '19 at 0:44
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    $\begingroup$ Also $\frac {dy}{dx} \frac {dx}{dt} = \frac {dy}{dt}$ looks natural and even obvious. Meanwhile $y'_x x'_t = y'_t$ looks arbitrary. $\endgroup$ – Džuris Dec 27 '19 at 0:46

A good way to start is setting clear that

$$ \frac{d}{dx}$$

as a block, is an operator, and

Operators are things that act on their right side.

So $\frac{d}{dx} y$ means "derivative of what is at the right", i.e. derivative of $y$.

I suggest you to highlight that sentence in yellow.

So $y \ \frac{d}{dx}$ just makes no sense because it is "$y$ times derivative of... nothing", as there's nothing at the right side.

I also like starting with $\frac{d}{dx}$ and not $\frac{dy}{dx}$, so that it is clearer that it is not a fraction, but a whole-block operator.

And finally, I'd suggest to read it as "derivative of", so that the students are tempted to complete the sentence in their minds. It's more serious than it seems haha.

Hope this helped.

  • $\begingroup$ Operators are things that act on their right side. --- Nearly always, yes. But not necessarily. (A) Students who use HP calculators with reverse Polish notation (pretty much all the serious math/physics/engineering students when I was in college) will disagree. (B) Herstein's Topics in Algebra, arguably the best known advanced undergraduate level abstract algebra (continued) $\endgroup$ – Dave L Renfro Dec 8 '18 at 16:41
  • $\begingroup$ textbook in the last 50 years, uses the reverse notation: (from p. 11, lines 5-7) "If $t$ is the image of $s$ under [the mapping] $\sigma$ we shall sometimes write this as ${\sigma}:s \rightarrow t;$ more often, we shall represent this fact by $t = s{\sigma}.$" Also, Herstein's book is not the only abstract algebra text on my shelves that uses this notation. $\endgroup$ – Dave L Renfro Dec 8 '18 at 16:41
  • $\begingroup$ Hm, that was really interesting, but extremely inneficient for a human being (in my view) though. Anyways, this was a practical rule for students, nothing rigorous, as you can see haha. OF course, if the students are using another notation would have to use the other. It's just about being coherent. $\endgroup$ – FGSUZ Dec 9 '18 at 14:04
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    $\begingroup$ An easy counterexample to "operators act on their right side" is the "prime" in $(x^2)' = 2x$. $\endgroup$ – Chris Cunningham Dec 10 '18 at 16:07
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    $\begingroup$ @ChrisCunningham: And so is exponentiation, such as the squaring notation. So your example above in fact is two examples at once. :-) $\endgroup$ – zipirovich Dec 25 '19 at 5:59

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