Simplify: $\;$ $\dfrac{1 - a^{\frac{-1}{2}}}{1 + a^{\frac{1}{2}}} - \dfrac{a^{\frac{1}{2}} - a^{\frac{-1}{2}}}{a - 1}$ $\;$ and calculate it for $\;$ $a = 5$.
$\dfrac{1 - a^{\frac{-1}{2}}}{1 + a^{\frac{1}{2}}} - \dfrac{a^{\frac{1}{2}} - a^{\frac{-1}{2}}}{a - 1}$
$= \dfrac{1 - \dfrac{1}{\sqrt{a}}}{\sqrt{a} + 1} - \dfrac{\sqrt{a} - \dfrac{1}{\sqrt{a}}}{a - 1}$
$= \dfrac{\sqrt{a} - 1}{\sqrt{a} \left(\sqrt{a} + 1\right)} - \dfrac{a - 1}{\sqrt{a} \left(a - 1\right)}$
$= \dfrac{\sqrt{a} - 1}{\sqrt{a} \left(\sqrt{a} + 1\right)} - \dfrac{1}{\sqrt{a}}$
$= \dfrac{\sqrt{a} - 1 - \left(\sqrt{a} + 1\right)}{\sqrt{a} \left(\sqrt{a} + 1\right)}$
$= \dfrac{-2}{a + \sqrt{a}}$
When $\;$ $a = 5$, $\;$ the given expression becomes
$\dfrac{-2}{5 + \sqrt{5}}$
$= \dfrac{-2 \left(5 - \sqrt{5}\right)}{\left(5 + \sqrt{5}\right) \left(5 - \sqrt{5}\right)}$
$= \dfrac{-2 \left(5 - \sqrt{5}\right)}{20}$
$= \dfrac{\sqrt{5} - 5}{10}$