you know that
d/du tan-1(u) = 1/(1+u^2)
So, use the chain rule and you have
u = 2x/(1-x^2)
du/dx = 2(1+x^2)/(1-x^2)^2
so, the derivative is
1/(1+(2x/(1-x^2))^2) * 2(1+x^2)/(1-x^2)^2 = 2/(1+x^2)
to see why this simplifies no neatly, consider that if
x = tanθ
2x/(1-x^2) = tan2θ
arctan(2x/(1-x^2)) = arctan(tan(2θ)) = 2θ = 2arctan(x)
the derivative is now 2/(1+x^2)
differentiate tan-1(2x/1-x^2)
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