Question

Prove that:
$\frac{d}{dx}\sqrt{\frac{1+x}{1-\sqrt{x}}}$

Answer 1

$\frac{d}{dx}\left(\sqrt{\frac{1+x}{1-\sqrt{x}}}\right)$

Using chain rule of differentiation, we have

$=\frac{1}{2\sqrt{\frac{1+x}{1-\sqrt{x}}}}\frac{d}{dx}\left(\frac{1+x}{1-\sqrt{x}}\right)$

$=\frac{1}{2\sqrt{\frac{1+x}{1-\sqrt{x}}}}\times \frac{(1-\sqrt{x})\frac{d}{dx}(1+x)-(1+x)\frac{d}{dx}(1-\sqrt{x})}{{(1-\sqrt{x})}^2}$

$=\frac{\sqrt{1-\sqrt{x}}}{2\sqrt{1+x}}\times \left[\frac{(1-\sqrt{x})-(1+x)\left(\frac{-1}{2\sqrt{x}}\right)}{{(1-\sqrt{x})}^2}\right]$

$=\frac{\sqrt{1-\sqrt{x}}}{2\sqrt{1+x}}\times \left[\frac{2\sqrt{x}-2x+1+x}{2\sqrt{x}{(1-\sqrt{x})}^2}\right]$

$=\frac{\sqrt{1-\sqrt{x}}}{2\sqrt{1+x}}\times \left[\frac{2\sqrt{x}-x+1}{2\sqrt{x}{(1-\sqrt{x})}^2}\right]$

$=\frac{2\sqrt{x}-x+1}{4\sqrt{x}\sqrt{1+x}(1-\sqrt{x})\sqrt{(1-\sqrt{x})}}$