Answer
$$\frac{x}{{{{\left( {{x^2} - 1} \right)}^2}}}$$
Work Step by Step
$$\eqalign{
& differentiate \cr
& = \frac{d}{{dx}}\left( { - \frac{1}{{2\left( {{x^2} - 1} \right)}} + C} \right) \cr
& = - \frac{1}{2}\frac{d}{{dx}}\left( {{{\left( {{x^2} - 1} \right)}^{ - 1}}} \right) + \frac{d}{{dx}}\left( C \right) \cr
& {\text{by the chain rule}} \cr
& = - \frac{1}{2}\left( { - 1} \right){\left( {{x^2} - 1} \right)^{ - 2}}\frac{d}{{dx}}\left( {{x^2} - 1} \right) + \frac{d}{{dx}}\left( C \right) \cr
& = - \frac{1}{2}\left( { - 1} \right){\left( {{x^2} - 1} \right)^{ - 2}}\left( {2x} \right) \cr
& {\text{simplify}} \cr
& = {\left( {{x^2} - 1} \right)^{ - 2}}\left( x \right) \cr
& = \frac{x}{{{{\left( {{x^2} - 1} \right)}^2}}} \cr} $$
