Given function and given differential of the function - how do I get there?

• November 7th 2006, 11:18 AM
sh01by
Given function and given differential of the function - how do I get there?
Hey mathematicians,

I have looked at this for quite some time...

I have (which is given)

f(x) = 2*x^2 / (x^4 +1)

and I have (which is also given)

f ' (x) = ( 4x (1 + x^2) (1+x) (1-x) ) / (x^4 + 1)^2

... So, I'm looking to find the steps to get to the given f ' (x)...

I have already worked quite a lot on it, but what i end getting is this...

I won't get into the (x^4 + 1)^2-part - as it is quite simple, but I get...

- 4x^5 + 4x => -4x (x^4 - 1)

Okay, so how do I get the last expression into factors?

What bothers me is the minus in front of 4x..? How do I get rid of it?

I hope someone has the time to help me out... It would be much appreciated.

Yours sincerely, Simon, DK.
• November 7th 2006, 11:31 AM
CaptainBlack
-4x (x^4 - 1) = -4x (x^2-1)(x^2+1) - using difference of two squares x^4 and 1

-4x (x^4 - 1) = -4x (x-1)(x+1)(x^2+1) - difference of two squares again x^2 and 1 in this case

-4x (x^4 - 1) = 4x (1-x)(x+1)(x^2+1) - using -(x-1)=(1-x).

RonL
• November 7th 2006, 11:42 AM
Soroban
Hello, Simon!

Quote:

$-4x^5 + 4x \;= \;-4x(x^4 - 1)$

What bothers me is the minus in front of $4x$ . . . How do I get rid of it?

Multiply through by it:

. . $-4x(x^4-1) \;=\;4x(-x^4 + 1) \;=\;4x(1 - x^4)$

• November 7th 2006, 11:43 AM
sh01by
Thank you!
Thank you very much. Quick reply. Perfect.