Derivative of a quotient inside a square root
You have\begin{align}-\frac1{\sqrt{x+1}\sqrt{(x-1)^3}}&=-\frac1{\sqrt{x+1}\sqrt{(x-1)^2}\sqrt{x-1}}\\&=-\frac1{\sqrt{(x+1)(x-1)}|x-1|}\\&=-\frac1{\sqrt{x^2-1}|x-1|}.\end{align}
Hints:
- $|a|=\sqrt{a^2}$
- $\sqrt{ab}=\sqrt a \sqrt b$
- $(a+b)(a-b)=a^2-b^2$.