The big argument: Is dy/dx notation better than y' or ẋ?

22 November 2021

Yes: dy/dx all the way, argues Ellen Jolley

Do you take me for an engineer, sir? The only acceptable notation for a derivative is the original notation created by Leibniz: $\mathrm{d}y/\mathrm{d}x$. This is the only notation that demonstrates precisely what the derivative is: the limit of the change in $y$ over the change in $x$ as the change in $x$ tends to zero. I will not stand idly by as the integrity and beauty of the derivative is lost in a sea of dashes and dots.

Anyone who insists on such primitive notation clearly could not have ventured far in their study of calculus: any student of multivariable calculus can see plainly that these pathetic diacritics are simply not up to the task. A simple extension to Leibniz’s notation allows me to write with ease any order of partial derivative with respect to any number of variables I choose, meanwhile the dashing-dotting hooligans are left scrambling.

High schoolers can also reap the benefits of Leibniz’s original notation: they may initially be bemused by the concept of a fraction that is not a fraction—but as soon as they study integration by substitution, off and away they go, writing $\mathrm{d}u = 2x\,\mathrm{d}x$ and so on. What exactly are we to do with $u’$? Prime-root it? And I suppose $\dot{u}$’s reciprocal is $\begin{smallmatrix}\displaystyle u\\ \cdot \end{smallmatrix}$?

No: dy/dget in the bin, argues Sophie Maclean

As mathematicians, we love dealing with fundamental truths and as truths go, ‘humans are lazy’ is about as fundamental as it gets. Mathematicians, as a subset of humans (it’s true—I looked it up) must therefore also be lazy. And we can see empirical evidence of this—I once sat staring at a problem for an hour trying to work out which method of solving it required the least writing. I’m pretty confident the phrase ‘work smart, not hard’ was invented for mathematicians.

So then why would anyone ever write $\mathrm{d}y/\mathrm{d}x$?! The effort it takes when compared to $\dot{x}$ is vast. Furthermore this will occur repeatedly throughout a paper! If you thought writing $\mathrm{d}y/\mathrm{d}x$ out by hand is slow, wait until you try typing it in LaTeX. I should point out here that I’m not the one typesetting this argument, hence I have no qualms about repeatedly writing $\mathrm{d}y/\mathrm{d}x$. $\mathrm{d}y/\mathrm{d}x$, $\mathrm{d}y/\mathrm{d}x$, $\mathrm{d}y/mathrm\{d\}x$.

It’s also so much quicker to read $\dot{x}$ than $\mathrm{d}y/\mathrm{d}x$. I’m all about the marginal gains, but in this case the gains are on an astronomical scale! And then you get on to the environmental impact. Wasting paper space writing $\mathrm{d}y/\mathrm{d}x$, when $\dot{x}$ does exactly the same job, is frankly not justifiable. What would Greta say?

Ellen Jolley

Ellen is a PhD student at UCL studying fluid mechanics. She specialises in the flow around droplets and ice particles.

Sophie Maclean

Sophie Maclean is a recent maths graduate from the University of Cambridge and very much misses her degree. She has no free time—she is a Chalkdust editor.