Find the average rate of change of $f$ from $\pi$ to $\frac{4 \pi}{3}$.
\[
f(x)=\tan x
\]

The average rate of change is $\square$.
(Simplify your answer, including any radicals. Type an exact answer, using $\pi$ as needed.)

Step 1 ：The average rate of change of a function \(f(x)\) from \(a\) to \(b\) is given by the formula: \(\frac{f(b) - f(a)}{b - a}\)

Step 2 ：Here, \(f(x) = \tan x\), \(a = \pi\), and \(b = \frac{4\pi}{3}\)

Step 3 ：We first need to find \(f(a)\) and \(f(b)\)

Step 4 ：\(f(a) = \tan(\pi) = 0\) (since tangent of any multiple of \(\pi\) is 0)

Step 5 ：\(f(b) = \tan\left(\frac{4\pi}{3}\right) = \tan\left(\pi + \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3}\) (since tangent of \(\pi/3\) is \(\sqrt{3}\))

Step 6 ：Substitute these values into the formula for the average rate of change: \(\frac{f(b) - f(a)}{b - a} = \frac{\sqrt{3} - 0}{\frac{4\pi}{3} - \pi} = \frac{\sqrt{3}}{\frac{\pi}{3}} = \frac{3\sqrt{3}}{\pi}\)

Step 7 ：So, the average rate of change of \(f\) from \(\pi\) to \(\frac{4 \pi}{3}\) is \(\boxed{\frac{3\sqrt{3}}{\pi}}\)