detailed working for differentiation

Hi, I've been working on my differentiation unit for the last couple of days and I am having a lot of trouble getting the correct answer.

as an example, for the question

$\displaystyle (dy)/(dx)$ of $\displaystyle y= 2/(x^2+1)$

i get an answer of

$\displaystyle (-4x)/(x^4+2x^3+x^2+2x+1)$

as apposed to the textbook answer of

$\displaystyle (-4x)/(x^4+2x^2+1)$

I've been looking through my working to find the problem, and i've found the location where i go in a different direction from the proper answer, but i can't see how i would change anything to get the right answer. Since i can't find the right answer with my own working out, my question is could somebody please point out where i go wrong and correct it (preferably in detail.

here is my working out (numerator is unimportant so i am not showing the working for it)

$\displaystyle =(2/((x+h)^2+1))-(2(x^2+1)$

$\displaystyle =((2(x^2+1))/((x^2+1)(x+h)^2+1))-((2((x+h^2)+1))/((x^2+1)(x+h)^2+1))$

$\displaystyle =(-4xh+-2h^2)/((x^2+1)(x+h)^2+1))$

the next 2 lines is where i suspect my mistake to be

$\displaystyle =(-4x+-2h)/(x^2(x^2+2hx+h^2)+1(x^2+2hx+h^2))$

$\displaystyle =(-4x+-2h)/(x^4+2hx^3+x^2h^2+x^2+2hx+h^2+1)$

lim h→0 $\displaystyle (-4x)/(x^4+2x^3+x^2+2x+1)$

$\displaystyle =(-4x)/(x^4+2x^3+x^2+2x+1)$

which does not $\displaystyle =(-4x)/(x^4+2x^2+1)$

Any help that is given would be greatly appreciated.