LaTeX Forum

\begin{document}
\usepackage{booktabs}
\begin{tabular}{cccc}
\toprule
\textsc{K}& \textsc{$\alpha$} & \textsc{$f^{\left(\alpha\right)}\left(x,y,z\right)$}& \textsc{$f^{\left(\alpha\right)}\left(a_1,a_2,a_3\right)$}  \\
\midrule
0 & & $f$ & $f(a_1,a_2,a_3)$ \\ 
\midrule
1& $(1,0,0)$ & $\frac{\partial f}{\partial x_1}$ & $\frac{\partial f}{\partial x_1}(a_1,a_2,a_3)$ \\
\cmidrule{2-4} & $(0,1,0)$ & $\frac{\partial f}{\partial x_2}$ & $\frac{\partial f}{\partial x_2}(a_1,a_2,a_3)$ \\
\cmidrule{2-4}& $(0,0,1)$ & $\frac{\partial f}{\partial x_3}$ & $\frac{\partial f}{\partial x_3}(a_1,a_2,a_3)$ \\
\midrule
2 & (2,0,0)  & $\frac{\partial^2 f}{\partial {x_1}^2}$ & $\frac{\partial^2 f}{\partial {x_1}^2}(a_1,a_2,a_3)$  \\
\cmidrule{2-4}  & (0,2,0)  & $\frac{\partial^2 f}{\partial {x_2}^2}$ & $\frac{\partial^2 f}{\partial {x_2}^2}(a_1,a_2,a_3)$  \\ 
\cmidrule{2-4} & (0,0,2)  & $\frac{\partial^2 f}{\partial {x_3}^2}$ & $\frac{\partial^2 f}{\partial {x_3}^2}(a_1,a_2,a_3)$  \\ 
\cmidrule{2-4}& (1,1,0)  & $\frac{\partial^2 f}{\partial {x_1}\partial{x_2}}$ & $\frac{\partial^2 f}{\partial {x_1}\partial{x_2}}(a_1,a_2,a_3)$  \\ 
\cmidrule{2-4}& (1,0,1)  & $\frac{\partial^2 f}{\partial {x_1}\partial{x_3}}$ & $\frac{\partial^2 f}{\partial {x_1}\partial{x_3}}(a_1,a_2,a_3)$  \\ 
\cmidrule{2-4}& (0,1,1)  & $\frac{\partial^2 f}{\partial {x_2}\partial{x_3}}$ & $\frac{\partial^2 f}{\partial {x_2}\partial{x_3}}(a_1,a_2,a_3)$  \\ 
\bottomrule
\end{tabular}

\end{document}

\documentclass{report}
\usepackage{booktabs, tabu} 
\begin{document} 
\extrarowsep=1mm


$\begin{tabu} to .7\textwidth {X[-1] X X[$,2,r]X[$,2,r]} 
\toprule 
\textsc{K}& $\alpha$ & f^{\left(\alpha\right)}\left(x,y,z\right)& f^{\left(\alpha\right)}\left(a_1,a_2,a_3\right)  \\ 
\midrule 
 0 & & f & f(a_1,a_2,a_3) \\ 
\midrule 
 1& (1,0,0) & \frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_1}(a_1,a_2,a_3) \\ 
 & (0,1,0) & \frac{\partial f}{\partial x_2} & \frac{\partial f}{\partial x_2}(a_1,a_2,a_3) \\ 
& (0,0,1) & \frac{\partial f}{\partial x_3} & \frac{\partial f}{\partial x_3}(a_1,a_2,a_3) \\ 
\midrule 
 2 & (2,0,0)  & \frac{\partial^2 f}{\partial {x_1}^2} & \frac{\partial^2 f}{\partial {x_1}^2}(a_1,a_2,a_3)  \\ 
  & (0,2,0)  & \frac{\partial^2 f}{\partial {x_2}^2} & \frac{\partial^2 f}{\partial {x_2}^2}(a_1,a_2,a_3)  \\ 
 & (0,0,2)  & \frac{\partial^2 f}{\partial {x_3}^2} & \frac{\partial^2 f}{\partial {x_3}^2}(a_1,a_2,a_3)  \\ 
& (1,1,0)  & \frac{\partial^2 f}{\partial {x_1}\partial{x_2}} & \frac{\partial^2 f}{\partial {x_1}\partial{x_2}}(a_1,a_2,a_3)  \\ 
& (1,0,1)  & \frac{\partial^2 f}{\partial {x_1}\partial{x_3}} & \frac{\partial^2 f}{\partial {x_1}\partial{x_3}}(a_1,a_2,a_3)  \\ 
& (0,1,1)  & \frac{\partial^2 f}{\partial {x_2}\partial{x_3}} & \frac{\partial^2 f}{\partial {x_2}\partial{x_3}}(a_1,a_2,a_3)  \\ 
\bottomrule 
\end{tabu}$

\end{document}