# [For experts] Derivatives of 1/f(x)^2

My question is presented in the uploaded pdf file.

:surprised

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But You might want to know that:
$$\frac{d}{dx}(\frac{1}{f^2(x)})=-2\frac{f'(x)}{f^3(x)}$$

I just briefly looked at the problem, but I wanted to say that the file is fine. Just a math problem. :)

Hurkyl
Staff Emeritus
Gold Member
I have a nitpick -- smooth functions (i.e. infinitely differentiable) are not required to have a MacLauren series -- you need to be analytic.

It may help

Thank you, Mr. Benorin. I'm trying to adapt the Faá di Bruno's formula to my problem. Bob

benorin
Homework Helper
Gold Member
Ok, so I found another formulation of Faa di Bruno's formula for the nth derivative of a composition of functions: here's your answer

$$\frac{d^{n}}{dx^{n}}\left(-\frac{1}{f^{2}(x)}\right) = \sum_{m=1}^{n}\left\{\frac{1}{m!}\left[\sum_{j=0}^{m-1}(-1)^{j}\frac{m!}{j!(m-j)!}f^{j}(x)\frac{d^{n}}{dx^{n}}\left( f^{m-j}(x)\right)\right]\frac{(-1)^{m+1}(m+1)!}{f^{m+2}(x)}\right\}$$

where $$f^{k}(x)$$ is the kth power of f(x) (not the kth derivative.)

-Ben

Mr. Benorin, you see, this is a local problem: the final result is evaluated at $$x = a$$. Besides that, $$f$$ satisfies some particular conditions, which must be considered:
(a) $$f(x) \neq 0$$, over some open interval $$A$$;
(b) $$f$$ is a series of even powers;
(c) $$f^{(2n+1)}(a) = 0$$ and $$f^{(2n)}(a) \neq 0$$, $$n = 0, 1, 2, ...$$;
The final result is a function of $$a$$, and the sum symbol, $$\Sigma$$, will not appear in the final answer.
As I've pointed,
$$g^{(2n)}(a)=-b_n f(a)^{-3n-2}$$​
Find $$(b_n)$$​
$$(b_n) = (1, 22, 584, 28384, 2190128, ...)$$​
I've encountered some difficulties to solve my task... 