I am trying to understand the chain rule under a change of variables. Given a function $f : \mathbb R^n \rightarrow \mathbb R$ and a change of variables $G : \mathbb R^m \rightarrow \mathbb R^n$, what is the derivative


$\partial^\alpha ( f \circ G )$


where $\alpha$ is a multiindex in the variables $x\_1,\dots,x\_m$ of degree $k$? We assume all necessary derivatives exist. References to the literature would be helpful too. I haven't found this general case treated in my sources.



Multivariable higher-order chain rule
You’re looking for the [multivariate version of the formula of Faa di Bruno](https://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula#Multivariate_version).


**Addendum**: As the OP notes, the version in Wikipedia is not in sufficient generality, since it takes $f:\mathbb R\to\mathbb R$. For a version that allows $f$ to depend on more than one variable, see for example the article:


* [A Multivariate Faa Di Bruno Formula with Applications, M. Constantine and T. H. Savits, *Trans. AMS*, **348**, Number 2, February 1996.](https://www.ams.org/journals/tran/1996-348-02/S0002-9947-96-01501-2/S0002-9947-96-01501-2.pdf)


