Solution
= Solution
Repeated use of the <product rule> and <chain rule> gives
$$
\boxed{f'(x)=\frac{-2x}{(1+x^2)^2},\quad f''(x)=\frac{2(3x^2-1)}{(1+x^2)^3},\quad f'''(x)=\frac{24x(1-x^2)}{(1+x^2)^4}.}
$$
The denominators are strictly positive on the whole real line, making the signs in the next parts easy to read.