# The fifth derivative of arctan(x)

1. Nov 29, 2012

### KiwiKid

1. The problem statement, all variables and given/known data
Show that the fifth derivative of arctan(x) equals (120x^4 - 240x^2 + 24)/((x^2 + 1)^5). Then, calculate the fifth order Taylor series of arctan(x) around x = 0.

2. Relevant equations
Differentiation rules; knowing how a Taylor series works.

3. The attempt at a solution
I tried the brute-force way of differrentiating those fractions over and over and over again, resulting in one big mess. My answer ultimately turned out to be something different than what was asked, which isn't particularly surprising, considering the number of calculations I had to perform. I didn't even get to the Taylor series part.

My question is: is there a non-obvious solution to this problem that I'm missing? If so, could anyone give me a hint? I've been stuck on this problem for over an hour now and I'm not exactly fond of the idea of redoing all my previous calculations.

2. Nov 29, 2012

### Ray Vickson

You need to show at least some of your work.

3. Nov 29, 2012

### KiwiKid

f(x) = arctan(x); The first derivative: f'(x) = 1/(x^2 + 1); Second derivative: f''(x) = -2x/((x^2+1)^2). ...And so on. Except that it's around the third derivative that it starts to get more complicated, and it becomes virtually impossible to keep differentiating without making mistakes. At least, that's what I think, because 'my' fifth derivative of arctan(x) didn't match what I was supposed to get at all.

4. Nov 29, 2012

### SammyS

Staff Emeritus
Writing those as:

f'(x) = (x^2 + 1)-1

and

f''(x) = -2x(x^2+1)-2

May help. You can then use the product rule rather than the quotient rule.

5. Nov 29, 2012

### KiwiKid

Wow, thanks. That's probably it. *starts writing it all down once more*

6. Nov 29, 2012

### Ray Vickson

What will also help is to put the whole thing over a common denominator before differentiating again, so before taking the (k+1)th derivative, make sure you write the kth derivative in the form
$$\frac{P_k(x)}{(x^2+1)^k},$$ where P_k(x) is a polynomial.