You could write this as
[tex]x(x^2-1)^{-1/2}[/tex]
Then by the product rule, the derivative is
[tex](x(x^2-1)^{-1/2})'=x((x^2-1)^{-1/2})'+(x)'(x^2-1)^{-1/2}=x\left(-\dfrac12(x^2-1)^{-3/2}(2x)\right)+(x^2-1)^{-1/2}[/tex]
[tex]\left(\dfrac x{\sqrt{x^2-1}}\right)'=-x^2(x^2-1)^{-3/2}+(x^2-1)^{-1/2}[/tex]
Pulling out a factor of [tex](x^2-1)^{-3/2}[/tex], the right hand side can be rewritten as
[tex]-(x^2-1)^{-3/2}\left(x^2-(x^2-1)^1\right)=-\dfrac1{(x^2-1)^{3/2}}[/tex]
