$y=\sqrt{x+1}.\dfrac{1}{\sqrt{x-1}}$
$=\dfrac{\sqrt{x+1}}{\sqrt{x-1}}$
$=\dfrac{\sqrt{x^2-1}}{x-1}$
$y'=\dfrac{(\sqrt{x^2-1})'(x-1)-\sqrt{x^2-1}(x-1)'}{(x-1)^2}$
$=\dfrac{\dfrac{(x^2-1)'}{2\sqrt{x^2-1}}.(x-1)-\sqrt{x^2-1}}{(x-1)^2}$
$=\dfrac{\dfrac{x}{\sqrt{x^2-1}}(x-1)-\sqrt{x^2-1}}{(x-1)^2}$
$=\dfrac{\dfrac{x\sqrt{x-1}}{\sqrt{x+1}}-\sqrt{x^2-1}}{(x-1)^2}$
$=\dfrac{x\sqrt{x-1}-\sqrt{(x^2-1)(x+1)}}{(x-1)^2\sqrt{x+1}}$
