Should students be asked to use more than one notation for the derivative in an introductory calculus class?

Question

There are many, many ways of writing the derivative of a function $y=f(x)$:

$$\frac{d}{dx}y, \frac{dy}{dx},\frac{d}{dx}f(x), \frac{df}{dx}, \dot y, D_x f,f',y',f'(x),f_x$$

and so on.

Students often feel uncomfortable switching back and forth between the various forms. And in fact, it is not necessary to use more than one form of notation in most beginning calculus courses.

However, knowing more than one form of notation may be helpful in future classes.

Should an instructor encourage their students to use different forms of derivative in their writing?

Answer 1

The different notations are to a considerable extent outgrowth of slightly different ways of looking at "the derivative." To me these different ways is the important thing to discuss, the notation is a by product. And, I think it is quite useful to spend some time on discussing these different ways and to mention the different notations in that context. (However I understand this is not always possible, see the end.)

For example:

• the notation $f'$ and $\dot{y}$ stresses the (physical) idea of having some function that describes something (some process) and one does something to it to analyze/understand its properties.

• the notation $D_x f$ emphasizes the fact that one can think of the situation of having an operator (a function!) $D_x$ that assigns to each (under suitable conditions) function $f$ another function $D_xf$.

• the notation $\frac{df}{dx}$ emphasizes the definition as a limit of the quotients of differences.

• the notation $df(x)$, in particular when used like $f(x+h) = f(x) + df(x)h + r(h)$ stresses that the derivative at a some point is actually a linear function that just happens to be identifiable with its slope.

There would also be different ways to split things up and to motivate things.

If one does this the different notations come more naturally and one then can and I think should vary one's own usage and exercises a bit. Still, I would not insist or even much encourage students using different notations themselves. There is some clear merit in having them understand the different notations; for them being able to use different ones (fuently) this is much less clear to me. There is also a small risk that they would use them, but would use them not really 'idiomatically,' as the notations arein some sense not completely interchangable.

For example even an example in OP seems slightly of to me (though some place else different conventions might be in place): as far as I know, the dot-notation is mainly used by physicist and alike, and does typically not denote just some derivative, but writing $\dot{y}$ "means" derivative by time, which "would never" be denoted $x$ but $t$. [For me, as a pure mathematician, the above is strange to write, but I really think that if $y=xt^2$, then to some it is completely obvious that $\dot{y}=2xt$ and not $\dot{y}=t^2$.]

This answer assumes of course that one has time to do this discussion, and implicitly also that the students or a considerable part there of will actually see the higher dimensional theory for example. If this is not the case, I think I would not bother much with different notations. Pick one (based on your own preference or possibly informed by the predominant convention in whatever other field the students might study, if there is one) and stick to it, and just mention once that there are different notations, so that students are not completely surprised when the open some other book. However, if some student prefers to use a different one, accept this, too (as long it is clear and coherent).

Answer 2

Yes, students will encounter several forms of derivative notation, inside and outside of classes. Therefore, a calculus class should give them fluency in reading these.

Whether they stick to one notation in their writing is not a big deal. More importantly: the lectures and questions in calculus classes should use more than one notation.

Answer 3

I think it is important to introduce the $d/dx$, prime and, to a much lesser extent, dot notations for the following reasons:

• $\frac{d}{dx}$ makes it quite clear which variable is being differentiated. Therefore, it is quite clear that $\frac{d}{dx}(a+x^2)=2x$ whereas $\frac{d}{da}(a+x^2) = 1$. In both of these calculations I assume that the complementary variable is held fixed in the differentiation. What do these look like in the prime notation? Well, $$ (a+x^2)' = 2x $$ is what my students would probably write. Perhaps, on occassion $(a+x^2)'=1$ would be offered by the sarcastic outlier. To actually be clear, we have to include the parenthetical evaluation: $$ (a+x^2)'(x) = 2x \qquad \& \qquad (a+x^2)'(a) = 1. $$ Even so, some might be tempted to write $(a+x^2)'(a) = 2a$. Which is it? The prime notation has danger in this regard.
• the prime notation is efficient. I think this may be one of the greatest aspects: $$ (f+g)' = f'+g' \qquad (cf)'=cf' \qquad (fg)'=f'g+fg' $$ and $$ \left(\frac{f}{g} \right)' = \frac{f'g-fg'}{g^2} \qquad (f \circ g)'=(f' \circ g)g' $$ It does require more writing to get these across in the $d/dx$ notation. Although, I do prefer to include the argument for the chain rule $(f \circ g)'(x)=f'(g(x))g'(x)$.
• the $\dot{x}$ and $\ddot{x}$ are fun. I just like to say $x$-double-dot. It takes me back to physics and so it has a certain nostalgic value for me personally. Pragmatically, these dot formulas are very pretty for expressing kinetic energy formulas for Lagrangian mechanics. When you use $\dot{x}$ as a variable it is a lot nicer than $x'$ or $dx/dt$. In variational calculus, the $\dot{x}$ appears as an "independent" variable from $x$.

Answer 4

Short answer: I don't think that is a good idea to encourage students to use different notation on their own. But I would encourage the lecturer and the TAs to do so and tell the students different notations.

More detailed answer:

For many students, the topic itsself is hard enough. I don't think you should distract them from the most important things for notation.

But you can tell them, that there are different ways to write down things and the most important point is the notation should be consistent. For example, I've seen it several times that at the point of the definition, there were several methods mentioned to write down the new object (here: the derivative) and then in the lecture one of these notations was used, but on the exercices another notation was used (this is - as I later learned - due to the fact that someone else was makeing exercises, but he hasn't talked about notation with the professor). Students will learn that different people use different notation and every notation has some advantages and disadvantages. Of course, it could be that students are confused at the first point ("What does that strange symbol in the exercises mean?" - You can explain it and you can write more than a symbol in the first exercise where it appears. Later on it should be clear).

You should encourage the students to make up their minds what a good notation is to their understanding. And the students should use that notation consequently as long as they are satisfied. (This changes from time to time. I for example was a big fan of the $\frac{\partial}{\partial x}f$ until I took a class about partial differential equations where I discovered that my old notation was a nightmare. Than I changed my whole notation).

I think, it is important that students should be flexible and open-minded to notation: It can be that in every lecture there is a different notation, in books and later in papers, people use different notation (in bad ones, the notation chances from chapter to chapter). But one should never overemphasize the discussion of notation since there are more important things to teach. And you should not jump arround with notation (otherwise the notation gets the focus and students will ask you about the meaning of the symbols rather than about the content).

Remark: Most of my arguments given is not specific to the notation of derivatives, but also in a general context. Maybe the most easy examples where some students have problems with notation is the example where an equation should not be solves for $x$, but for a different variable. Another example would be the use of another symbol than $f$ for a function.

Answer 5

You only need $\frac{d}{dx}$, $\frac{dy}{dx}$, and $f^\prime$.

$\frac{d}{dx}\Psi$, where $\Psi$ is an expression whose value depends on that of $x$, is the rate of change of the value of $\Psi$ with respect to $x$.

$$\frac{d}{dx}\Psi = \lim_{h \rightarrow 0} \frac{f\left(x+h\right) - f\left(x\right)}{h}\ \text{where}\ f = k \mapsto \left(\Psi,\ \text{given}\ x=k\right)$$

$\frac{dy}{dx}$ is the rate of change of a dependent variable ($y$) with respect to an independent variable ($x$). Even though $y$ is simply an expression whose value depends on $x$ (making $\frac{dy}{dx}$ and $\frac{d}{dx}y$ synonyms), $\frac{dy}{dx}$ is nice because its form reminds us of the intuition that comes from $\frac{\Delta y}{\Delta x}$.

$$\frac{dy}{dx}= \lim_{h \rightarrow 0} \frac{f\left(x+h\right) - f\left(x\right)}{h}\ \text{where}\ f = k \mapsto \left(y,\ \text{given}\ x=k\right)$$

$f^\prime$, where $f$ is a function with one argument, is the rate of change of change of $f$'s output with respect to $f$'s input.

$$f^\prime = k \mapsto \lim_{h \rightarrow 0} \frac{f\left(k+h\right) - f\left(k\right)}{h}$$

Some caveats:

$\frac{df}{dx}$ and $\frac{d}{dx} f$ are meaningless because $f$ is a function, not a variable or an expression. The value of $f$ does not change with $x$; the value of $f\left(x\right)$ changes with $x$. So use $\frac{d}{dx} f\left(x\right)$ instead. Note that $\frac{d}{dx} f\left(x\right)$ is a synonym of $f^\prime\left(x\right)$. Also note that $\frac{d}{dx} f\left(a\right)=0$ because $f\left(a\right)$ is not an expression that depends on $x$.

Likewise, $y^\prime$ is meaningless because $y$ is a variable, not a function. $y$ could depend on any number of variables, so there's no unambiguous way to write $y$ as a function. But you could theoretically write: $\left( k \mapsto \left(y,\ \text{given}\ x=k \right)\right)^\prime\left(x\right)$ which is a synonym of $\frac{dy}{dx}$.

Answer 6

I'm for using different notations (particularly $\frac{\mathrm{d} f}{\mathrm{d} x}$ and $f'(x)$, as they are useful for emphasizing different aspects; and occasionally $\mathrm{D} f$ if the operator aspect is central). But don't overdo it, saying the same thing in five different ways "just because we can" will only confuse the reader. That they are expressions of different ways of seeing derivatives, from much before they were really understood, is incidental. And those views might even harm understanding, better leave them out until the concept is crystal clear and solid.

Answer 7

I think that the only possible answer is: it depends. Is fluency with various notation a goal of the course? If yes, you have your answer, if no, you would better concentrate on other stuff. Dealing with notation is hard and can be confusing and time-consuming, so don't engage in it unless you have good reasons to.

As an example, for engineering students it might make sense to have your course using the notation they will find in other courses, or to emphasize a notation which is more suited to your topics and a notation they will find elsewhere, to help them relate what you teach and the context in which they will have use it.

Answer 8

A thought from Steven Krantz's How to Teach Mathematics, 3rd Ed., published by the American Mathematical Society (sec. 2.12):

If a textbook uses notation or other conventions that you do not like, then don't use that book. You really are obliged to follow the notation and definitions and other paradigms in the text you have chosen. Otherwise all but the gifted students will be lost. If you repeatedly criticize the text as the course proceeds, then you will be sending a confusing message to the students: Why did you choose this book if it is obviously so full of flaws? Isn't it your job to select a text that you can teach from?

Granted that a textbook is unlikely to vary between multiple different notations, then the recommendation here is likely "no".

Answer 9

I think some reasonable fluency is needed. But I would mostly stick with dy/dx and y'.

I would also avoid using the most obscure (least used) notation when you are in the middle of introducing a new derivation or something complicated (e.g. integration by parts). Try to stay with something familiar in those cases (probably dy/dx).

Answer 10

If one ponders seriously the question "If $d/dx$ is an operator, on what does it operate?", one comes to the conclusion that it doesn't operate on functions in the modern sense of the word (i.e. objects that map elements from a subset of $\mathbb{R}$ to elements of $\mathbb{R}$). Instead, it operates on objects of a different kind, for which the historically correct name is functions of $x$.

(Warning: although we use these other objects all the time, it seems that they were never formalized in modern mainstream mathematics. So we are in the awkward situation of teaching something which we can't really make precise. But I'll just pretend that we all know what a functions of $x$ is.)

If we compare the prime and the $d/dx$ notations from this perspective, we see that they are not two different notations for the same thing, but operators on two different spaces. Eulers prime operates on (modern) functions, while Leibniz' $d/dx$ operates on functions of $x$.

This is the convention I currently use when I teach. So for example I would never write $(x^2+1)'$, since $x^2+1$ is not a function (in the modern sense), but a function of $x$. Instead I write $\frac{d}{dx}(x^2+1)$. (There are other reasons for avoiding the prime notation when differentiating functions of $x$, like the impossibility to do the chain rule with it.). Similarly, if $f$ is a function like $x\mapsto (x^2+1)$ I would never write $\frac{df}{dx}$ since $f$ is not a function of $x$. Instead I write $f'$. On the other hand, if $g$ is a function depending on a parameter, like $x\mapsto x^2+c$, then it does make sense to write $\frac{dg}{dc}$, but this is not the same as $g'$.