Multivariate version

Let ${\displaystyle y=g(x_{1},\dots ,x_{n})}$. Then the following identity holds regardless of whether the ${\displaystyle n}$ 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]

${\displaystyle {\partial ^{n} \over \partial x_{1}\cdots \partial x_{n}}f(y)=\sum _{\pi \in \Pi }f^{(\left|\pi \right|)}(y)\cdot \prod _{B\in \pi }{\partial ^{\left|B\right|}y \over \prod _{j\in B}\partial x_{j}}}$

where (as above)

• ${\displaystyle \pi }$ runs through the set ${\displaystyle \Pi }$ of all partitions of the set ${\displaystyle \{1,\ldots ,n\}}$,
• "${\displaystyle B\in \pi }$" means the variable ${\displaystyle B}$ runs through the list of all of the "blocks" of the partition ${\displaystyle \pi }$, and
• ${\displaystyle |A|}$ denotes the cardinality of the set ${\displaystyle A}$ (so that ${\displaystyle |\pi |}$ is the number of blocks in the partition ${\displaystyle \pi }$ and

${\displaystyle |B|}$ is the size of the block ${\displaystyle B}$).

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 ${\displaystyle \{1,2,3\}}$, and in each case the order of the derivative of ${\displaystyle f}$ is the number of parts in the partition:

${\displaystyle {\begin{aligned}{\partial ^{3} \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}f(y)={}&f'(y){\partial ^{3}y \over \partial x_{1}\,\partial x_{2}\,\partial x_{3}}\\[10pt]&{}+f''(y)\left({\partial y \over \partial x_{1}}\cdot {\partial ^{2}y \over \partial x_{2}\,\partial x_{3}}+{\partial y \over \partial x_{2}}\cdot {\partial ^{2}y \over \partial x_{1}\,\partial x_{3}}+{\partial y \over \partial x_{3}}\cdot {\partial ^{2}y \over \partial x_{1}\,\partial x_{2}}\right)\\[10pt]&{}+f'''(y){\partial y \over \partial x_{1}}\cdot {\partial y \over \partial x_{2}}\cdot {\partial y \over \partial x_{3}}.\end{aligned}}}$

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.

Formal power series version

Suppose ${\displaystyle f(x)=\sum _{n=0}^{\infty }{a_{n}}x^{n}}$ and ${\displaystyle g(x)=\sum _{n=0}^{\infty }{b_{n}}x^{n}}$ are formal power series and ${\displaystyle b_{0}=0}$.

Then the composition ${\displaystyle f\circ g}$ is again a formal power series,

${\displaystyle f(g(x))=\sum _{n=0}^{\infty }{c_{n}}x^{n},}$

where ${\displaystyle c_{0}=a_{0}}$ and the other coefficient ${\displaystyle c_{n}}$ for ${\displaystyle n\geq 1}$ can be expressed as a sum over compositions of ${\displaystyle n}$ or as an equivalent sum over integer partitions of ${\displaystyle n}$:

${\displaystyle c_{n}=\sum _{\mathbf {i} \in {\mathcal {C}}_{n}}a_{k}b_{i_{1}}b_{i_{2}}\cdots b_{i_{k}},}$

where

${\displaystyle {\mathcal {C}}_{n}=\{(i_{1},i_{2},\dots ,i_{k})\,:\ 1\leq k\leq n,\ i_{1}+i_{2}+\cdots +i_{k}=n\}}$

is the set of compositions of ${\displaystyle n}$ with ${\displaystyle k}$ denoting the number of parts,

or

${\displaystyle c_{n}=\sum _{k=1}^{n}a_{k}\sum _{\mathbf {\pi } \in {\mathcal {P}}_{n,k}}{\binom {k}{\pi _{1},\pi _{2},\ldots ,\pi _{n}}}b_{1}^{\pi _{1}}b_{2}^{\pi _{2}}\cdots b_{n}^{\pi _{n}},}$

where

${\displaystyle {\mathcal {P}}_{n,k}=\{(\pi _{1},\pi _{2},\dots ,\pi _{n})\,:\ \pi _{1}+\pi _{2}+\cdots +\pi _{n}=k,\ \pi _{1}\cdot 1+\pi _{2}\cdot 2+\cdots +\pi _{n}\cdot n=n\}}$

is the set of partitions of ${\displaystyle n}$ into ${\displaystyle k}$ parts, in frequency-of-parts form.

The first form is obtained by picking out the coefficient of ${\displaystyle x^{n}}$ in ${\displaystyle (b_{1}x+b_{2}x^{2}+\cdots )^{k}}$ "by inspection", and the second form is then obtained by collecting like terms, or alternatively, by applying the multinomial theorem.

The special case ${\displaystyle f(x)=e^{x}}$, ${\displaystyle g(x)=\sum _{n\geq 1}{\frac {1}{n!}}a_{n}x^{n}}$ gives the exponential formula. The special case ${\displaystyle f(x)=1/(1-x)}$, ${\displaystyle g(x)=\sum _{n\geq 1}(-a_{n})x^{n}}$ gives an expression for the reciprocal of the formal power series ${\displaystyle \sum _{n\geq 0}a_{n}x^{n}}$ in the case ${\displaystyle a_{0}=1}$.

Stanley ^[4] gives a version for exponential power series. In the formal power series

${\displaystyle f(x)=\sum _{n}{\frac {a_{n}}{n!}}x^{n},}$

we have the ${\displaystyle n}$th derivative at 0:

${\displaystyle f^{(n)}(0)=a_{n}.}$

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

${\displaystyle g(x)=\sum _{n=0}^{\infty }{\frac {b_{n}}{n!}}x^{n}}$

and

${\displaystyle f(x)=\sum _{n=1}^{\infty }{\frac {a_{n}}{n!}}x^{n}}$

and

${\displaystyle g(f(x))=h(x)=\sum _{n=0}^{\infty }{\frac {c_{n}}{n!}}x^{n},}$

then the coefficient ${\displaystyle c_{n}}$ (which would be the ${\displaystyle n}$th derivative of ${\displaystyle h}$ evaluated at 0 if we were dealing with convergent series rather than formal power series) is given by

${\displaystyle c_{n}=\sum _{\pi =\left\{B_{1},\ldots ,B_{k}\right\}}a_{\left|B_{1}\right|}\cdots a_{\left|B_{k}\right|}b_{k}}$

where ${\displaystyle \pi }$ runs through the set of all partitions of the set ${\displaystyle \{1,\ldots ,n\}}$ and ${\displaystyle B_{1},\ldots ,B_{k}}$ are the blocks of the partition ${\displaystyle \pi }$, and ${\displaystyle |B_{j}|}$ is the number of members of the ${\displaystyle j}$th block, for ${\displaystyle j=1,\ldots ,k}$.

This version of the formula is particularly well suited to the purposes of combinatorics.

We can also write with respect to the notation above

${\displaystyle g(f(x))=b_{0}+\sum _{n=1}^{\infty }{\frac {\sum _{k=1}^{n}b_{k}B_{n,k}(a_{1},\ldots ,a_{n-k+1})}{n!}}x^{n},}$

where ${\displaystyle B_{n,k}(a_{1},\ldots ,a_{n-k+1})}$ are Bell polynomials.

A special case

If ${\displaystyle f(x)=e^{x}}$, then all of the derivatives of ${\displaystyle f}$ are the same and are a factor common to every term:

${\displaystyle {d^{n} \over dx^{n}}e^{g(x)}=e^{g(x)}B_{n}\left(g'(x),g''(x),\dots ,g^{(n)}(x)\right),}$

where ${\displaystyle B_{n}(x)}$ is the nth complete exponential Bell polynomial.

In case ${\displaystyle g(x)}$ is a cumulant-generating function, then ${\displaystyle f(g(x))}$ is a moment-generating function, and the polynomial in various derivatives of ${\displaystyle g}$ is the polynomial that expresses the moments as functions of the cumulants.