Chapter 5 - Trigonometric Functions - Section 5.5 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions - 5.5 Assess Your Understanding - Page 442: 43

$\frac{6\sqrt 3}{\pi}$.

1. Given $f(x)=tan(2x)$, we have $f(0)=tan(0)=0$ and $f(\frac{\pi}{6})=tan(\frac{\pi}{3})=\sqrt 3$, 2. The rate of change required is given by $\frac{f(\frac{\pi}{6})-f(0)}{\frac{\pi}{6}-0}=\frac{\sqrt 3-0}{\frac{\pi}{6}}=\frac{6\sqrt 3}{\pi}$.