Does unit elasticity has to be at exactly the middle of the demand curve? Can unit elasticity be anywhere else on the demand curve other than the midpoint?
No, that is only true in the linear case. For a simple counterexample consider
$$
D(p) = 1 - \sqrt{p}.
$$
$$
\epsilon(p) = \frac{d D(p)}{dp} \cdot \frac{p}{D(p)} = -\frac{1}{2  \cdot \sqrt{p}} \cdot \frac{p}{1 - \sqrt{p}} = \frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}}
$$
\begin{eqnarray*}
\epsilon(p) &amp; = &amp; 1 \\
\\
\frac{1}{2}  \cdot \frac{\sqrt{p}}{1 - \sqrt{p}} &amp; = &amp; 1 \\
\\
\sqrt{p} &amp; = &amp; 2 - 2 \cdot \sqrt{p}  \\
\\
\sqrt{p} &amp; = &amp; \frac{2}{3} \\
\\
p &amp; = &amp; \frac{4}{9}.
\end{eqnarray*}
This is not in the middle of the demand curve in any sense.