<s>[INST] Write a Limit to calculate $f'(0)$ Let $f(x) = \frac {2}{1+x^2} $ 
I need to write a limit to calculate $f'(0)$.
I think I have the basic understanding. Any help would be greatly appreciated.
d=delta and so far what I have is 
$f'(x)$= lim           (f(x+dx)-f(x))/dx
    (dx)->0
((2/1+(x+dx)^2)-(2/1+x^2))/dx
((2/1+x^2+2xdx+dx^2)-(2/1+x^2))/dx
((2(1+x^2)-2(1+x^2+2xdx+dx^2))/(1+x^2+2xdx+dx^2)(1+x^2))/dx
((-4xdx-2dx^2)/(1+x^2+2xdx+dx^2)(1+x^2))/dx
(-4xdx-2dx^2)/(1+x^2+2xdx+dx^2)(1+x^2)(dx)
(-2dx(2xdx+dx)/(1+x^2+2xdx+x^2)(1+x^2)(dx)
(-2(2xdx+dx)/(1+x^2+2xdx+dx^2)(1+x^2)
that's as far as I have gotten. Any input would be great.
 [/INST] it is $$\frac{f(x+h)-f(x)}{h}=\frac{\frac{2}{1+(x+h)^2}-\frac{2}{1+x^2}}{h}$$
Can you finish?
Ok, another hint:
The numerator is given by $$-2\,{\frac {h \left( h+2\,x \right) }{ \left( {h}^{2}+2\,xh+{x}^{2}+1
 \right)  \left( {x}^{2}+1 \right) }}
$$
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