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How would one analytically write the following;

$$\frac{d^n}{dx^n}\left(f(x)^k\right)$$

for integer $k$ and integer $n$? Writing it for a specific $n$ is easy, however, is there a finite series (similar to the Stirling or Bell numbers) that will generate a correct solution?

Gabriel Walentin
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  • Interesting problem. I know that $$ \frac{{\rm d}^n}{{\rm d}x^n} x^k = \frac{k!}{(k-n)!} x^{k-n}$$ but the problem above is more general. – John Alexiou May 24 '17 at 20:22

1 Answers1

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Faa di Bruno's formula gives the $n$-th derivative of $g(f(x))$ so just take $g(y)=y^k$ therein.

Angina Seng
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  • Is there any advantage using Faa di Bruno's formula over using the one for higher products (https://en.wikipedia.org/wiki/Product_rule#Higher_derivatives)? – Sudix Aug 29 '19 at 04:30