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Is there a closed form formula for $$\frac{d^n}{d t^n}F(x(t),y(t))$$ in terms of partial derivatives of $F$? I have worked out the expression partially: $$\frac{d^n}{d t^n}F(x(t),y(t))=\sum_{i=1}^{n}\sum_{j=0}^i f_{j,i-j}F^{(j,i-j)}(x(t),y(t))$$ and have formulas for $f_{1,0}$, $f_{0,1}$, $f_{2,0}$, $f_{0,2}$, $f_{1,1}$, but I don't see a way to generalize to the other coefficients without concatenating more sums, which I would like to avoid.

Prastt
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  • There is an analogous general chain rule for total derivatives which might be relevant to your generalization: http://en.wikipedia.org/wiki/Fa%C3%A0_di_Bruno%27s_formula. – jdc May 23 '14 at 19:18
  • @jdc, thanks for the suggestion, I think I might have found the generalization in this paper http://www.emis.de/journals/HOA/IJMMS/Volume24_7/498526.pdf I have to check it in detail. – Prastt May 23 '14 at 19:58

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