Homework Help: [For experts] Derivatives of 1/f(x)^2

I am not sure I want to download the file... sorry.
But You might want to know that:
[tex]\frac{d}{dx}(\frac{1}{f^2(x)})=-2\frac{f'(x)}{f^3(x)}[/tex]

 

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

 

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

 

Use Faá di Bruno's Formula for the Nth derivative of a composition of functions (since 1/f(x)^2=h(f(x)) where h(x)=1/x^2). Here is a link:

http://mathworld.wolfram.com/FaadiBrunosFormula.html

hope you like integer partitions...

 

Ok, so I found another formulation of Faa di Bruno's formula for the n^th derivative of a composition of functions: here's your answer

[tex]\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\}[/tex]

where [tex]f^{k}(x)[/tex] is the k^th power of f(x) (not the k^th derivative.)

-Ben