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.

  • Can you give the context? Exact differentials? Also is yd/dx (second form you list) the same as dy/dx. – guest Dec 3 at 22:52
  • @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. – Ben Crowell Dec 3 at 22:57
  • 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. – Rusty Core Dec 4 at 0:38
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    @TommiBrander: This is a community college in the US, i.e., they're in their first or second year of university, at a school with nonselective admissions. – Ben Crowell Dec 5 at 2:17
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    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. – Dave L Renfro Dec 5 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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    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.[...] – Ben Crowell Dec 5 at 2:12
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    [...] 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. – Ben Crowell Dec 5 at 2:14
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    @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. – Benoît Kloeckner Dec 5 at 11:19
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    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 $x^2$?". – Michael Bächtold Dec 6 at 22:01
  • @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. – Dave L Renfro Dec 7 at 12:19

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:

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

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.

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  • 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) – Dave L Renfro 2 days ago
  • 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. – Dave L Renfro 2 days ago
  • 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. – FGSUZ yesterday
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    An easy counterexample to "operators act on their right side" is the "prime" in $(x^2)' = 2x$. – Chris Cunningham 20 hours ago

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