# Statisfaction

## Faà di Bruno’s note on eponymous formula, trilingual version

Posted in General by Julyan Arbel on 20 December 2016

The Italian mathematician Francesco Faà di Bruno was born in Alessandria (Piedmont, Italy) in 1825 and died in Turin in 1888. At the time of his birth, Piedmont used to be part of the Kingdom of Sardinia, led by the Dukes of Savoy. Italy was then unified in 1861, and the Kingdom of Sardinia became the Kingdom of Italy, of which Turin was declared the first capital. At that time, Piedmontese used to commonly speak both Italian and French.

Faà di Bruno is probably best known today for the eponymous formula which generalizes the derivative of a composition of two functions, $\phi\circ \psi$, to any order:

$(\phi\circ \psi)^{(n)} = \sum \frac{n!}{m_1!\,\ldots m_n!}\phi^{(m_1+\,\cdots \,+m_n)}\circ \psi \cdot \prod_{i=1}^n\left(\frac{\psi^{(j)}}{j!}\right)^{m_j}$

over $n$-tuples $(m_1,\,\ldots \,, m_n)$ satisfying $\sum_{j=1}^{n}j m_j = n.$

Faà di Bruno published his formula in two notes:

• Faà Di Bruno, F. (1855). Sullo sviluppo delle funzioni. Annali di Scienze Matematiche e Fisiche, 6:479–480. Google Books link.
• Faà Di Bruno, F. (1857). Note sur une nouvelle formule de calcul différentiel. Quarterly Journal of Pure and Applied Mathematics, 1:359–360. Google Books link.

They both date from December 1855, and were signed in Paris. They are similar and essentially state the formula without a proof. I have arXived a note which contains a translation from the French version to English (reproduced below), as well as the two original notes in French and in Italian. I’ve used for this the Erasmus MMXVI font, thanks Xian for sharing!

NOTE ON A NEW FORMULA FOR DIFFERENTIAL CALCULUS.

By M. Faà di Bruno.

Having observed, when dealing with series development of functions, that there did not exist to date any proper formula dedicated to readily calculate the derivative of any order of a function of function without resorting to computing all preceding derivatives, I thought that it would be very useful to look for it. The formula which I found is well simple; and I hope it shall become of general use henceforth.

Let $\phi(x)$ be any function of the variable $x$, linked to another one $y$ by the equation of the form

$x=\Psi(y).$

Denote by $\Pi(l)$ the product $1\, . \, 2\, . \,3\, \ldots \,l$ and by $\Psi^\prime,\,\Psi^{\prime\prime},\,\Psi^{\prime\prime\prime},$ etc the successive derivatives of the function $\Psi(y)$; the value of $D_y^n\phi$ will have the following expression:

$D_y^n\phi = \sum\frac{\Pi(n)}{\Pi(i)\Pi(j)\ldots\Pi(k)} D_x^p\phi\cdot \left(\frac{\Psi^{\prime}}{1}\right)^i \left(\frac{\Psi^{\prime\prime}}{1\, . \,2}\right)^j \left(\frac{\Psi^{\prime\prime\prime}}{1\, . \,2\, .\,3}\right)^h\ldots\,\left(\frac{\Psi^{(l)}}{\Pi(l)}\right)^k.$

the sign $\sum$ runs over all integer and non negative values of $i,j,h\ldots k$, for which

$i+2j+3h\ldots +lk = n,$

the value of $p$ being

$p=i+j+h\,\ldots\,+k$

The expression can also take the form of a determinant, and one has

It is implicit that the exponents of $\phi$ will be considered as orders of derivation.

Paris, December 17. 1855.