Who invented the Leibnitz notation $\frac{d^2y}{dx^2}$ for the *second* derivative?

Question

This MSE question made me wonder where the Leibnitz notation $\frac{d^2y}{dx^2}$ for the second derivative comes from. It does not arise immediately as the obvious generalization of $\frac{dy}{dx}$. Did Leibnitz use it himself? Or was it introduced later?

Answer 1

Leibniz did use this notation for instance in his paper Supplementum geometriae practicae, Acta Eruditorum, April 1693, p. 179 (Google Books link):

Answer 2

The differential symbold $dx$ is due to Leibniz.

He introduced also "iterated" differentials; see :

• H.J.M. Bos, Differentials and Higher-Order Differentials in the Leibnizian Calculus (1974), page 17:

Moreover, to introduce higher-order differentials, first-order differentials have to be conceived as variables ranging over an ordered sequence; if only a single $dx$ is considered, $ddx$ does not make sense. The following quotation from Leibniz ["Monitum de characteribus algebraicis", 1710] illustrates this:

Further, $ddx$ is the element of the element or the difference of the differences, for the quantity $dx$ itself is not always constant, but usually increases or decreases continually.

See also The Early Mathematical Manuscripts of Leibniz (J.M. Child ed., 1920 - also Dover reprint): manuscript of an answer to Bernhard Nieuwentijt, page 144-on:

$dx, ddx, dv, ddv, dy, ddy \ldots$

We have to note that Leibniz has $xx$ for $x^2$; see page 151 :

Then, since $y-xx : a \ldots$

Answer 3

The accepted answer leaves no doubt that Leibniz was the first to write $d^2y/(dx)^2$ for the second derivative. But since I've found so many misleading justifications for this notation online, I feel that something additional needs to be said about it.

Most justifications in the links above are along the lines of: "by formal manipulation" or "too obviously" $$ \frac{d}{dx} \frac{d}{dx} =\frac{d^2}{dx^2}. \tag1 $$ But Leibniz, the Bernoullis or Euler would not have approved of this without reservation. Not even if the equation was written in the form $$ \frac{d \left(\frac{dy}{dx} \right)}{dx} = \frac{d dy}{(dx)^2}, \tag2 $$ which is closer to the standard of the time.

To explain let me make a simple analogy first. No one today would claim that the following is correct $$ \frac{\log \frac{\log y}{\log x}}{\log x} = \frac{\log \log y}{(\log x)^2}, \tag3 $$ and everyone can spot the error.

Analogously, for Leibniz, $d$ was an operator (he might not have called it that way, but he knew it acted on variables just like $\log$) and he knew the quotient rule for $d$. So he might have approved of the following general equation

$$ \frac{d \left(\frac{dy}{dx} \right)}{dx} = \frac{d^2y}{(dx)^2} - \frac{dy\cdot d^2x}{(dx)^3}. \tag4 $$ The reason the second term on the right disappeared, was because an additional assumption was often made: it was assumed that the differential of the differential of $x$ is zero (i.e. $d^2x=0$), or put differently: $dx$ was assumed constant.

This can be seen in the 1693 article of Leibniz quoted by @ViktorBlasjo, a line above $ddx:\overline{dy}^2$, where he writes

posita $dy$ constante

It can also be found in Eulers Institutiones Calculi Differentialis (1743) § 131.

Now we will proceed under the assumption that $x$ increases uniformly, so that the first differentials $dx, dx^I , dx^{II},\ldots$ are equal to each other, so that the second and higher differentials are equal to zero. We can state this condition by saying that the differential of $x$, that is $dx$, is assumed to be constant. Let $y$ be any function of $x$; ...

And it can be found in Lacroix's Traité du calcul différentiel et du calcul intégral (1797) p.96

Pour la simplifier nous observons que l'accroissement $dx$ étant regardé invariable, $f'(x)dx$ se change en $f'(x+dx)dx$ ...

Summarizing: for Leibniz, Euler and others the equation $$ \frac{d \left(\frac{dy}{dx} \right)}{dx} = \frac{d^2y}{dx^2} \tag5 $$ was only true under the additional assumption that $dx$ is constant.

This leaves a question for me, which hopefully someone else can answer: when and why did mathematicians forget this additional assumption and simply adopt the notation $\frac{d^2}{dx^2}$ for what should actually be written as $\left(\frac{d}{dx}\right)^2$?

Answer 4

I think $\frac{d^2y}{dx^2}$ comes from multiplying $\frac{dy}{dx}$ by $\frac{d}{dx}$. In the Notation(https://en.m.wikipedia.org/wiki/Abuse_of_notation#Derivitive) multiplication signifies iteration.

(Disclaimer; This is a very rough response. There have not been any other answers yet, I will look for the notation in a textbook.)

Answer 5

$\frac{d^2}{dx^2}$ $y$ = $\frac{d^2y}{dx^2}$ is too obviously built from $\frac{d}{dx}$$\frac{d}{dx}$ $y$ = $(\frac{d}{dx})^2$ $y$ to deserve any further explanation.