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

  • Thread starter Thread starter eok20
  • Start date Start date
  • Tags Tags
    Derivative
Click For Summary
SUMMARY

The discussion centers on finding the 100th derivative of the function f(x) = x/(1+x^2) at the point x=1. Participants suggest using power series expansion and partial fraction decomposition to tackle the problem. The power series converges for |x| < 1, but the challenge arises at x=1. A key insight is to consider the series expansion around x=1, which may provide a clearer path to the solution.

PREREQUISITES
  • Understanding of power series and their convergence criteria
  • Familiarity with partial fraction decomposition
  • Knowledge of derivatives and their computation
  • Basic complex analysis concepts related to roots of polynomials
NEXT STEPS
  • Learn about power series expansion around specific points
  • Study partial fraction decomposition techniques in detail
  • Explore Taylor series and their applications in derivative calculations
  • Investigate convergence of series involving complex roots
USEFUL FOR

Mathematics students, educators, and anyone interested in advanced calculus, particularly those focusing on derivatives and series expansions.

eok20
Messages
198
Reaction score
0

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?
 
Physics news on Phys.org
Maybe I'm missing something but since this is a second-order equation, aren't all derivatives past the second one zero?
 
Fleem, it is a fraction.

Hint: split it in partial fractions.
 
Count Iblis said:
Fleem, it is a fraction.

Hint: split it in partial fractions.

I need glasses.
 
eok20 said:

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
 

Similar threads

Find f s. t. ||f||=1 and f(x) < 1 with ||x||=1
  • · Replies 20 ·
Replies
20
Views
3K
Find f(x,y) given partial derivative and initial condition
  • · Replies 6 ·
Replies
6
Views
2K
Differentiable function proof given ##f''(c) = 1##
  • · Replies 1 ·
Replies
1
Views
1K
If f is undefined at x=a, can f' exist at x=a?
  • · Replies 9 ·
Replies
9
Views
1K
Determine for which x the derivative exists for $$f(x)=arcsin(\sqrt x)$$
  • · Replies 16 ·
Replies
16
Views
2K
Prove that this series converges uniformly on R
  • · Replies 4 ·
Replies
4
Views
2K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
1K
How to show that these sequences are summable?
  • · Replies 1 ·
Replies
1
Views
2K
Find the constrained maxima and minima of ##f(x,y,z)=x+y^2+2z##
  • · Replies 2 ·
Replies
2
Views
2K
Derivative Problem: f’(1), f’(2), f’(3), and f’(5)
  • · Replies 11 ·
Replies
11
Views
2K