Is There an Easier Way to Find the Nth Derivative of Curvature?

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The discussion focuses on the complexity of finding the nth derivative of the curvature function \(\kappa(x) = \frac{f''(x)}{[1 + (f'(x))^2]^\frac{3}{2}}\). The user explores using Faa Di Bruno's formula in conjunction with the Leibniz formula, only to encounter recursive complications that necessitate repeated applications of these formulas. Ultimately, the consensus is that there is no simpler method available for this task, confirming the intricate nature of the problem.

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JPBenowitz
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So I am trying to find the nth derivative of the curvature function:

[itex]\kappa(x)[/itex] = [itex]\frac{f''(x)}{[1 + (f'(x))^2]^\frac{3}{2}}[/itex]

Now, I could go about using Faa Di Bruno's formula but when I did I realized that I also have to use the Leibniz formula as a substitution for a term in the Faa Di Bruno formula. Once I did that low and behold I had to use Faa Di Bruno's formula again for a term in Leibniz formula... so on and so forth about two times again. Is there any easier route to go about finding the nth derivative?
 
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Probably not. It is in itself a question with a messy answer.
 

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