Question about how Ashcroft and Mermin came to Equation 1.56

I’m trying—unsuccessfully, as of yet—to follow how the second half of equation 1.56 is derived in Solid State Physics by Ashcroft and Mermin (equations below). Please keep in mind that I’m using this inaccurate derivation of thermopower for a high school research essay, so I’d appreciate it if things could be explained as simply as possible.

Ashcroft and Mermin are essentially deriving an estimate of thermopower in the Drude model of conduction in a metal. As part of this, they derive an equation for the average velocity at a point in a metal bar under a temperature gradient, using v(x) to denote the average velocity of electrons at point x:

How is it that $frac{1}{2}[v(x – vtau) – v(x + vtau)] = -tau v frac{dv}{dx}$?