An ideal p - n junction diode has a current-voltage characteristic...

The current-voltage characteristic of the diode is given by $i = i_0 (e^{eV/kT} - 1)$.

For $V > V_0$, the term $e^{eV/kT} \ge 100$, so $1$ can be neglected compared to $e^{eV/kT}$.

The expression for current becomes $i \approx i_0 e^{eV/kT}$.

The dynamic resistance $r_d$ of a diode is defined as $r_d = \dfrac{dV}{di}$.

Differentiating the current with respect to voltage $V$, we get:

$\dfrac{di}{dV} = \dfrac{d}{dV} (i_0 e^{eV/kT}) = \dfrac{e i_0}{kT} e^{eV/kT}$

Therefore, the dynamic resistance is:

$r_d = \dfrac{1}{\dfrac{di}{dV}} = \dfrac{kT}{e i_0} e^{-eV/kT}$