What is the derivative of this function:

$\displaystyle abs(\frac{x}{1+x^2})$

This is the shot I took:

$\displaystyle \frac{x}{1+x^2}$ and $\displaystyle \frac{-x}{1+x^2}$

so $\displaystyle \frac{(x-1)(x-1)}{(x^2+1)^2}$ and $\displaystyle \frac{(x+1)(x+1)}{(x^2+1)^2}$ would be the derivative of those functions respectively

But I really don't understand the fundamentals of this problem - the absolute value is really throwing me and I don't know how to properly deal with it. Thanks so much if you can help!