information:filtering, linguistic:voice_change, structural:reformatting, structural:expansion, linguistic:paraphrasing
Faà di Bruno's formula

If f and g are n -times differentiable, then d n d x n = n ! ∑ { k m } f (r) (g (x)) ∏ m = 1 n 1 k m ! (g (m) (x)) k m {\displaystyle {\frac {d^{n}}{dx^{n}}}=n!\sum _{\{k_{m}\}}f^{(r)}(g(x))\prod _{m=1}^{n}{\frac {1}{k_{m}!}}\left(g^{(m)}(x)\right)^{k_{m}}} where r = ∑ m = 1 n − 1 k m {\textstyle r=\sum _{m=1}^{n-1}k_{m}} and the set { k m } {\displaystyle \{k_{m}\}} consists of all non-negative integer solutions of the Diophantine equation ∑ m = 1 n m k m = n {\textstyle \sum _{m=1}^{n}mk_{m}=n}.