We owe a debt to Bell Labs with Shockley, Bardeen and Brattain for inventing the semiconductor. All have now passed on. I remember seeing my first transistor radio in the early 60's. Shockley developed the very important diode equation. I was reading through one of my good old engineering books where the dynamic resistance of a diode is derived. I'm trying to understand this because it's basic to many applications for transistors, current mirrors, long tailed pairs, etc. The dynamic resistance is used to get the volt drop over the small signal ac resistance. They give the equation $$I =I_s \left( e^{eV/kT}-1 \right) \tag{1}$$ where $I$ is the diode current, $V$ is the diode voltage, $k$ is Boltzmann's constant, $T$ is the temperature in Kelvin, and $I_s$ is the reverse leakage current.
They then derive the dynamic resistance as $dV/dI$ and get an answer as roughly $1/40I$.
I can't get the same answer that they do. Instead, I have to use this equation from Wikipedia to get the same answer: $$I=I_s \left( e^{V/V_t}-1 \right) \tag{2}$$ where $V_t$ is the thermal volts, which is 25mV at room temperature. Are they using the wrong equation in (1), or have I worked it out wrong?