Answer
\begin{array}{|c|c|c|c|c|c|} \hline x & {-4} & {-2} & {0} & {2} & {4} & {8} \\ \hline y & {2} & {0} & {4} & {4} & {6} & {8} \\
\hline \frac{d y }{ d x} & {\sqrt 3} & {0} & -{\sqrt 3} &- {\sqrt 3} & {0 } & {\sqrt 3} \\ \\ \hline
\end{array}
Work Step by Step
Given$$\dfrac{dy}{dx}= \tan \frac{\pi y}{6 }$$ and
\begin{array}{|c|c|c|c|c|c|}\hline x & {-4} & {-2} & {0} & {2} & {4} & {8} \\ \hline y & {2} & {0} & {4} & {4} & {6} & {8} \\ \hline \dfrac{d y }{d x } & { } & { } & { } & { } & { } & { } \\ \hline\end{array}
$$$$
Since the $1^{st}$ derivative $(\dfrac{dy}{dx}) $ mean the slope, so by substituting by the values of points $(x,y)$, we get
\begin{array}{|c|c|c|c|c|c|} \hline x & {-4} & {-2} & {0} & {2} & {4} & {8} \\ \hline y & {2} & {0} & {4} & {4} & {6} & {8} \\ \hline \frac{d y }{ d x} & {\tan \frac{ \pi }{3 }} & { \tan 0 }& {\tan \frac{2 \pi }{3 }} & {\tan \frac{2 \pi }{3 }} & {\tan \pi } & {\tan \frac{4\pi }{3 } } \\ \\ \hline
\hline \frac{d y }{ d x} & {\sqrt 3} & {0} & -{\sqrt 3} &- {\sqrt 3} & {0 } & {\sqrt 3} \\ \\ \hline
\end{array}