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

\(\displaystyle f'(x)=(x^2+1)d/dx(-x)-(-x)d/dx(x^2+1)/(x^2+1)^2\)

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

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

Is what I came up with

Hold on, you said \(\displaystyle f(x) = -\frac{1}{x^2+1}\) in the OP. Which one is correct?

Your differentiating is correct but your simplification on the third line is not - there should be no x term. Plus be more clear with your syntax using brackets

\(\displaystyle u = -x \: \rightarrow \: u' = -1\)

\(\displaystyle v = x^2+1 \: \rightarrow \: v' = 2x\)

\(\displaystyle f'(x) = \frac{-x^2-1 + 2x^2}{(x^2+1)^2} = \frac{x^2-1}{(x^2+1)^2}\)