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Joe Silverman
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You’re looking for the multivariate version of the formula of Faa di Bruno.

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:

You’re looking for the multivariate version of the formula of Faa di Bruno.

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:

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Zach Teitler
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Joe Silverman
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