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I am trying to get the pattern of the Faa di Bruno's formula of the chain rule for higher derivatives. The only thing which I don't understand is how to get the coefficients of the various terms.

For example, let $y=g(x)$. Now $(f\circ g)''''(x)=f''''(y)y'^4+6f'''(y)y''y'^2+3f''(y)y''^2+4f''(y)y'''y'+f'(y)g''''$.

Here we note that that each term corresponds with a partition of $4$. The first term has four $y'$ terms in it and so it corresponds with $1+1+1+1$ (this sum has four terms and hence fourth derivative of $f(y)$ is there). The second term has a $y''$ and two $y'$ terms in it and so corresponds with $2+1+1$ (this sum has 3 terms and hence third derivative of $f(y)$ is there) and so on. But how can the pattern of the coefficients be expressed now?

Thanks

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    http://mathworld.wolfram.com/FaadiBrunosFormula.html2012-12-09
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    One may wonder why you did not google this! :-D The two answers so far are a link to mathworld and to the Wikipedia page on the subject, both of which Google is quite good at finding.2013-03-24

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