Homework Help: Simplifying a Derivative

1. Jul 16, 2012

communitycoll

1. The problem statement, all variables and given/known data
Find the derivative of arctan[(1 - x) / (1 + x)].

2. Relevant equations
Everything in the "Show Steps" section:
http://www.wolframalpha.com/input/?i=derivative+arctan[(1+-+x)+/+(1+++x)]

My problem is that I don't know how Wolfram manages to simplify:

- 2 / [1 + ((1 - x) / (1 + x))^2](1 + x)^2

^ which is also what I've managed to get,

to get ,

- 1 / (1+x^2)

3. The attempt at a solution
Everything you see Wolfram does pretty much.

2. Jul 16, 2012

Muphrid

It multiplies through by the (1+x)^2 on the right and then expands the squares.

But really, it might be more instructive to use the chain rule and do this by hand?

3. Jul 16, 2012

genericusrnme

try multiplying everything out in the denominator and see what happens

4. Jul 16, 2012

alan2

Just multiply out your denominator. You get -2/{(1+x)^2+(1-x)^2}=-2/{2+2x^2}=....

5. Jul 16, 2012

communitycoll

Tell me what to do next or what I've done wrong here:

just showing the denominator:

= 1 + 2x + x^2 + [(1 - 2x + x^2)(1 + 2x + x^2)(1 + 2x + x^2) / (1 + 2x + x^2)]

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

= 1 + 2x + x^2 + 1 + 2x + x^2 - 2x - 4x^2 - 2x^3 + x^2 + 2x^3 + x^4

= 2 + x^4 - 2x^2 + 2x

6. Jul 16, 2012

genericusrnme

The denominator you gave was
[1 + ((1 - x) / (1 + x))^2](1 + x)^2

take the (1+x)^2 inside the brackets
[(1 + x)^2 + (1 + x)^2((1 - x) / (1 + x))^2]

Go from there, I'm not sure how you started off with
"= 1 + 2x + x^2 + [(1 - 2x + x^2)(1 + 2x + x^2)(1 + 2x + x^2) / (1 + 2x + x^2)]"

7. Jul 16, 2012

communitycoll

Ah, never mind, I was thinking I'm meant to multiply the fraction in the denominator as if they were being added/subtracted, trying to multiply the numerators by the denominators of both the fraction and (1 + x)^2. I get it now. Thanks, I appreciate your patience : D

Last edited: Jul 16, 2012