A notation moves no theorem, and it does move which mistake looks natural.
This paper sets side by side, in one computation, the places where Leibniz’s notation keeps its promise and the places where it breaks it. NO NEW MATHEMATICAL THEOREM IS CLAIMED: the triple product relation, the fact that the second derivative is not a fraction, Leibniz’s product rule, Faa di Bruno’s formula with the Bell numbers, and the subscript on a partial derivative are all standard. Nor does this paper rule on the historical priority dispute, since only the behaviour of the notation is treated.
First, a premise. Notation does not move the answer. The derivative of the sine of x squared times the exponential of minus x, taken at 0.7, is 0.37970904324369351073 whether computed numerically or from the closed form, the two agreeing to ten to the minus thirty-two. Newton would write a dot and Leibniz a quotient, and the number is the same. A notation is a convention and moves no theorem. Everything below stands on that premise.
Second, and this is the core. The Leibniz quotient looks like a fraction, and in the one-variable chain rule it really does behave like cancellation, which is how that success gets carried into the next situation. But once three variables are tied by one relation the answer changes. For the ideal gas with the gas constant set to one, computing the three partial derivatives with the third variable held fixed gives minus 0.75, 0.6666666666666667 and 2.0, whose product is minus one. This is no quirk of the ideal gas: on the implicit surface given by the sum of the cubes plus the product equal to ten, the triple product is minus one at all three sample points computed. Cancelling gives minus one every time, because the notation invites the cancellation and the cancellation drops a sign. Newton’s dot cannot invite it, having no denominator. The two notations write the same theorems and differ in which mistakes are available: what a convention selects is not truth but slipperiness.
Third, at second order the fraction reading collapses. With one variable the cube of a parameter and the other its square, the true second derivative at parameter values one, two and three is minus 0.22222222222222, minus 0.013888888888889 and minus 0.0027434842249657, while reading it as a ratio of second derivatives gives plus 0.33333333333333, plus 0.16666666666667 and plus 0.11111111111111. Not merely a different value, but a different sign: the true expression is minus two over nine times the fourth power, and the ratio reading gives plus one over three times the parameter. The second-derivative symbol is one symbol, not a fraction of two things, and being right at first order guaranteed nothing at second.
Fourth, the same notation keeps one promise and breaks another. The product rule really is the binomial theorem: with an exponential and a sine at 0.4, the derivative computed directly and the binomial sum agree at orders one, two, four and six, giving 6.567908305582929, minus 0.6941515037403181, minus 343.6136864015332 and 3553.448468147446. The number of terms is one more than the order and their weights total two to the power of the order. The chain rule does not keep it. With the sine composed with an exponential minus one, at 0.3, Faa di Bruno’s formula at fourth order reproduces minus 15.82942435520884 exactly, but the multiplicities used are one, then four and three, then six, then one, totalling fifteen, which is the fourth Bell number. Setting the two growths side by side, at order ten the product rule has eleven terms of total weight 1024 and the chain rule has forty-two terms of total weight 115975. One notation takes different attitudes to two neighbouring rules, so the claim that the differential may be used like a fraction can only be settled rule by rule. What comes out of a notation is a conjecture, not a proof.
Fifth, one symbol plays two roles. Let a quantity be the sum of two squares with the second variable equal to three times the first. Both of the following are called the derivative with respect to the first variable, yet holding the second fixed gives 2.0, 4.0 and 10.0 at the points one, two and five, while letting it follow gives 20.0, 40.0 and 100.0, a ratio of ten throughout. A factor of ten separates two things the bare notation writes the same way, which is why thermodynamics writes the held variable as a subscript. The subscript is not decoration: without it the expression denotes no number at all, the same shape as what Paper 16 wrote about the equals sign.
Finally, how the notation was eventually justified. Using the differential as an infinitesimal was for a long time an operation that worked without grounds. It was justified in Cartan’s differential forms, where the differential became a defined object, a one-form, and cancellation became a defined operation, pullback along a map, as Paper 92 describes. Leibniz’s notation, in other words, was given its meaning two centuries later. It kept working throughout, and working is not the same as meaning something: the limits of that working are the third, fourth and fifth points above. A census: the one-variable chain rule looks like cancelling and is correct; three tied variables look as if cancelling gives one and give minus one; the second-derivative symbol looks like a fraction and differs even in sign; the product rule looks binomial and is binomial; the higher chain rule looks simple and gives Bell numbers; a partial derivative is one symbol with two meanings.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
⚠️ Scope ── what is not claimed
a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem, none is claimed; a verdict on the priority dispute, none is given. What is added: setting out in one computation how a single notation behaves at five different places, and in particular putting the product rule’s weight and the chain rule’s Bell numbers into one table. What remains: a convention may be chosen freely, and what cannot be chosen is which mistake it invites. Leibniz’s notation has supplied correct guesses for three centuries and, over the same three centuries, has made people slip in the same places, and where they slip was settled the moment the notation was chosen.
Reference
Yamagishi, Y. (2026). Cancelling the Differentials Gives $-1$, Not $+1$ [Paper 170]. Zenodo.
DOI: 10.5281/zenodo.22093731
ORCID: 0009-0006-8587-3450