Question

Simplify the expression for $f\left(x\right)$ and then find $f^{\prime }\left(x\right)$, if
$f\left(x\right)={(\frac{\sqrt{x-2}}{\sqrt{x+2}+\sqrt{x-2}}+\frac{x-2}{\sqrt{x^2-4}-x+2})}^{-2}(\frac{x-1}{2(\sqrt{x}+1)}+1)\times \frac{2}{\sqrt{x}+1}$

Answer 1

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

$f\left(x\right)={(\frac{\sqrt{x-2}}{\sqrt{x+2}+\sqrt{x-2}}+\frac{\sqrt{x-2}}{\sqrt{x+2}-\sqrt{x-2}})}^{-2}$

$f\left(x\right)={\left(\frac{2\sqrt{x^2-4}}{4}\right)}^{-2}$

$f\left(x\right)=\frac{4}{x^2-4}$

$f^{\prime }\left(x\right)=\frac{0-4\left(2x\right)}{(x^2-4{)}^2}=\frac{8x}{(x^2-4{)}^2}$