$y=\sqrt{x+1}.\Big(\dfrac{1}{\sqrt{x}}-1\Big)$
$=\dfrac{ \sqrt{x+1} }{\sqrt{x}} - \sqrt{x+1}$
$y'= \dfrac{(\sqrt{x+1})'\sqrt{x}-\sqrt{x+1}(\sqrt{x})' }{x} - \dfrac{1}{2\sqrt{x+1}}$
$=\dfrac{ \dfrac{1}{2\sqrt{x+1}}.\sqrt{x} - \sqrt{x+1}.\dfrac{1}{2\sqrt{x}} }{x} -\dfrac{1}{2\sqrt{x+1}}$
$=\dfrac{ \sqrt{x}.2\sqrt{x}-\sqrt{x+1}.2\sqrt{x+1} }{x.4\sqrt{x}.\sqrt{x+1}} -\dfrac{1}{2\sqrt{x+1}}$
$=\dfrac{ 2x-2x-2 }{4x\sqrt{x}\sqrt{x+1}} -\dfrac{1}{2\sqrt{x+1}}$
$=\dfrac{-1}{2x\sqrt{x}\sqrt{x+1}} -\dfrac{1}{2\sqrt{x+1}}$
$=\dfrac{-x\sqrt{x}-1 }{2\sqrt{x}\sqrt{x+1}}$
