Question

Differentiate the function w.r.t.x:
$\rm \frac{\sqrt{x}+1}{\sqrt{x}-1}$

Answer 1

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Answer 2

To solve the given differentiation,we have to apply U/V formula

Now apply U/V formula of differentiation

$\frac{d}{dx} ( \frac{U}{V} ) = \frac{V\frac{dU}{dx} - U \frac{dV}{dx} }{ {V}^{2} } \\ \\ here \: U = \sqrt{x}+ 1 \\ \\ V = \sqrt{x}- 1$
$\frac{d}{dx} ( \frac{\sqrt{x}+ 1 }{\sqrt{x}- 1} ) = \frac{(\sqrt{x} - 1)\frac{d(\sqrt{x}+ 1) }{dx} - (\sqrt{x}+ 1) \frac{d\sqrt{x}- 1}{dx} }{ {(\sqrt{x} - 1)}^{2} }$
$= \frac{(\sqrt{x} - 1)\frac{1}{2\sqrt{x}} - (\sqrt{x}+ 1) \frac{1}{2\sqrt{x}}}{ {(\sqrt{x} - 1)}^{2} }$
$=\frac{(\sqrt{x}-1-\sqrt{x} -1)}{2\sqrt{x}{(\sqrt{x} - 1)}^{2}}$
$=\frac{- 2}{2\sqrt{x}{(\sqrt{x} - 1)}^{2}}\\ \\\frac{d}{dx} \bigg( \frac{\sqrt{x}+ 1 }{\sqrt{x}- 1} \bigg) =\frac{-1}{\sqrt{x}{(\sqrt{x} - 1)}^{2}}$
Hope it helps you.