Sorry for my longtime absence on this platform. Was just thinking of you all the other day and thought I’d post. Instead of my normal ID content, I thought I’d post another interesting area for discussion – the Leibniz notation for the second derivative. My contention is that it is wrong in a deep way, though there are enough kludges available to make the numbers work in normal situations, which is why we get correct answers while using the wrong notation.
In short, the Leibniz notation is a ratio of differentials. However, if you actually treated the second derivative as a ratio of differentials you would get the wrong answer. The response is normally “well, it’s just notation, it’s not a real ratio”. I could understand that perspective EXCEPT that there is an ACTUAL RATIO of differentials that you CAN treat directly as a ratio. As such, since there exists a ratio of differentials that works, and the Leibniz notation is not it, that means that the standard Leibniz notation is actually an incorrect ratio of differentials.
The standard notation for the second derivative is:
To clarify what this is supposed to mean, d^2y is basically d(d(y)) and dx^2 is basically (d(x))^2. However, it doesn’t actually work as a fraction. The actual formula for the second derivative which DOES work as a fraction is:
With this notation, differentials can be freely manipulated just as any other algebraic expression. This notation can be easily derived from applying the quotient rule to the first derivative. In other words, if you treat the first derivative as an actual quotient and just apply the rule, you get the notation.
The two papers covering this are:
- Bartlett and Khurshudyan, “Extending the Algebraic Manipulability of Differentials“, which covers the core concept, and
- Fite and Bartlett, “Total and Partial Differentials as Algebraically Manipulable Entities“
There have also been two math videos made on this subject. I’ve only seen the first one all the way through. I haven’t had time to watch the second one all the way yet. One drawback on the first video is that the explanation takes a shortcut that I think undermines what he is doing – he makes an assumption that d^2t = 0, but he only needed to do that because he was keeping the OLD notation for the left-hand side. If he just subtracted it from both sides (instead of assuming it goes to zero), he would have both a correct answer on the right-hand side and the new notation on the left-hand side.
One thing that almost no one mentions, which I think is the more interesting part, is that this allows for different ratios of differentials to be considered. I don’t know what practical application this would bring, but it means that some ideas may be easier to express than before. For instance, the below ratio of differentials is a technically valid ratio under the new system, but was essentially unexpressible in the old system (if you don’t see the difference, notice the placement of the 2 in the denominator):
Also helpful is that the improved notation helps connect together various aspects of scientific mathematics that is not always made clear, but can be shown to be algebraically true using this system.
Curious as to people’s thoughts.
Hi johnnyb — I agree that the standard Leibniz notation is kind of a mess, and not just because of the second derivative problem you’ve pointed out. It’s also confusing because dx looks like the product of d and x when d is actually an operator. A non-alphabetic character would have been a better choice, as in the example of Δx. Writing dx as d(x) would work, but at the expense of clunkiness. Also annoying is the fact that
looks like a fraction but is actually an operator.
Regarding the second derivative, you’re absoutely right that the second derivative of y(x) is not
Instead, it’s
My guess is that the latter didn’t become the standard notation because although it is strictly correct, it would be tiring to write it out all the time. More importantly, if x is an independent variable, which is the usual case, then
is 0 and the second term vanishes, at which point 
truly is equal to the second derivative.
To me, it looks like the tradeoff is between using the correct but clunkier expression versus using the simpler one but remembering that it can only be treated as a ratio when x is an independent variable. When x isn’t independent, the clunkier expression is mandatory if you want to manipulate it algebraically.
Speaking of second derivatives, you might be amused by an OP I did a year ago in which I asked various LLMs the simple question “What is the value of the second derivative at points of inflection?” I wanted to see if they’d recognize that the second derivative isn’t always zero at a point of inflection — it can also be undefined. The results were amusing:
An AI loses it
keiths,
“if x is an independent variable, which is the usual case, then d^2x is 0 and the second term vanishes”
Technically, this is only true if dx is *constant* (and I also wouldn’t say it’s the “usual case” except in contrived homework problems). The derivative being constant *may* be equivalent with being independent, but I think this conception may break down with more variables.
As for writing, in my own calculus book, I always used a non-italic “d” and an italic “x” just like you would “sin x”, but closer together, in order to typographically emphasize that “d” was the operator and “x” was the variable. I’m not sure anyone has noticed or cared, but I cared 🙂
I should point out that I also wrote a short paper on why a differential-focused calculus better prepares students for thinking mathematically, as well as how to lean into a differential-focused calculus for even more gain. Here’s a simple example. It’s trivial, but it is also pedagogically important. Most calculus books give the volume of rotation around the x-axis as:
We ask students to memorize this. However, the way it is written actually breaks apart the components. What we are integrating is a whole bunch of cyclinders. The formula for a cylinder is pi r^2 h. But that isn’t readily visible to the student in the formula, so it becomes just another “here is a formula to memorize” rather than really understanding that we are just adding lots of cylinders. So, in my book, this formula is written as:
This emphasizes that this isn’t just an arbitrary formula – it *came* from somewhere, and you can still see the traces of that in the formula.
johnnyb:
The two conditions aren’t equivalent, but one implies the other. If
is independent, then 
, but 
being zero doesn’t guarantee that 
is independent.
Another way to state this is that if
isn’t constant, then 
isn’t independent. Something outside of 
itself is causing 
to change, and whatever that thing is, 
depends on it.
If we ask a student to find the second derivative of
, we don’t ding them for giving the answer 
. They’re entitled to assume that 
is independent unless told otherwise. It’s not like indefinite integration, where we do ding students for leaving out the “+ C” additive constant.
I care, so that makes at least two of us. I applaud your attention to detail.
When I learned multivariable calculus, there were a lot of problems that involved computing the volumes of weird solids (or areas of weird surfaces) where constructing the integral correctly relied on the kind of geometric intuition you’re describing and no standard formulas were available. It’s really just a multivariate extension of the idea that the integral of a function can be thought of as the sum of infinitesimally wide rectangles under a curve. You figure out a way to add a bunch of small things together to approximate a larger one, and then you take the limit of the sum as the small things get smaller and smaller.
keiths,
Just as a note, the second derivative of x^3 is 6x whether or not x is independent. It’s the left-hand side that it matters for, unless I’m missing your meaning.
johnnyb:
You’re right. I was mistaking the second term on the RHS as an adjustment to the second derivative, when in fact it’s an adjustment to
in order to bring it in line with the second derivative. What I should have said is that if we ask the student for 
, we don’t ding them if their answer is 
. That answer is correct under the assumption that 
is independent, which is a reasonable assumption given that we haven’t specified otherwise.
To summarize: The second derivative really is
if x is independent, and it can legitimately be treated as a fraction in that case, which is the point I’m making. Otherwise it must be interpreted as an operator with a conventional but misleading notation. That’s the point your OP is making.
I just watched the first video, and the creator seems to have reached the same conclusion as I did. If x is independent, then
is truly equal to the second derivative and can be treated as a ratio. If x is dependent, then it can only be treated as notation, not as an actual ratio. You need johnnyb’s adjustment term, 
, if you want to treat it as a ratio.
The second video is almost two hours, and life is short.