Is the Notation for the Second Derivative Wrong?

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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:

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.

One thought on “Is the Notation for the Second Derivative Wrong?”

  1. 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 \frac{d}{dx} 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

        \[ y''(x) = \frac{d^2y}{dx^2} \]

    Instead, it’s

        \[ y''(x)= \frac{d^2y}{dx^2} - \frac{dy}{dx}\frac{d^2x}{dx^2} \]

    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 d^2x is 0 and the second term vanishes, at which point \frac{d^2y}{dx^2} 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

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