Calculating Higher Order Derivatives of a Complex Function

[CantorSet]

Hi everyone,

I came across this problem that I can't solve, but I'm sure there's a simple elegant solution.

Let $f(x) = \frac{x}{(1-(x^2))^2}$.

Find $f^{2011}(0)$.

 

[micromass]

Hi everyone,

I came across this problem that I can't solve, but I'm sure there's a simple elegant solution.

Let $f(x) = \frac{x}{(1-(x^2))^2}$.

Find $f^{2011}(0)$.

Begin by taking the first few derivatives and hope to see a pattern emerging. So, what are the first, second, third, etc. derivatives?

 

[Citan Uzuki]

I think you'll have better luck trying to construct the Taylor series expansion than trying to compute the derivatives manually. By Taylor's theorem, the coefficient of x^2011 will be f^(2011)(0)/2011!, so multiply it by 2011! to get your answer.

 

[CantorSet]

Thanks for responding everyone.

Micromass, I got up to the 3rd derivative but I unfortunately did not see any nice pattern. :(

Citan Uzuki, I see what you mean. So I basically got that

$\frac{x}{(1-(x^2))} = \sum_{n=1} n x^{2n-1}$

by playing around with the geometric series. Now, you're saying that this is also the Taylor Series expansion? In which case, the answer I get is 1005 * 2011!

See, I thought of this earlier, but I didn't realize that every nice function has a unique power series expansion. That is, the Taylor Series is also the geometric series.

 

[Citan Uzuki]

Yep, that's the approach exactly! Just one thing - for the exponent to be 2011, n must be 1006, not 1005. But otherwise, you got it exactly right.

 

[bpet]

Also x/(1-x^2)^2 = (1/2)(d/dx)(1/(1-x^2))