Find 100th Derivative of f(x) = x/(1+x^2) at 1

[eok20]

Homework Statement

Find the 100th derivative at 1 of f(x) = x/(1+x^2)

Homework Equations

The Attempt at a Solution

For |x| < 1 I can write f as a power series (since 1/(1+x^2) = sum_n (-1)^n x^(2n)) but this won't work at 1. I tried writing out the first few derivatives at 1 explicitly but things got pretty messy. Any ideas?

 

[fleem]

Maybe I'm missing something but since this is a second-order equation, aren't all derivatives past the second one zero?

 

[Count Iblis]

Fleem, it is a fraction.

Hint: split it in partial fractions.

 

[fleem]

Fleem, it is a fraction.

Hint: split it in partial fractions.

I need glasses.

 

[John Creighto]

The Attempt at a Solution

For |x| < 1 I can write f as a power series (since 1/(1+x^2) = sum_n (-1)^n x^(2n)) but this won't work at 1. I tried writing out the first few derivatives at 1 explicitly but things got pretty messy. Any ideas?

Are you just trying to find the derivative at one point? If so, and you are trying to find the derivative at the point x=1, then why not do the power series at about x=1. If you use partial fractions as another poster suggested, the roots will be +/=i so I think the series should converge