How to add extra spaces between rows in tabular environment?

You have at least three options here:

• Increasing the array stretch factor using \renewcommand{\arraystretch}{<factor>} where <factor> is a numeric value:

  \documentclass{article}
  \begin{document}
  \renewcommand{\arraystretch}{2}
  \begin{tabular}{c c}
    $f^{(n)}(x)$ & $f^{(n)}(0)$ \\\hline
    $-2xe^{-x^{x^{x^2}}}$ & 0
  \end{tabular}
  \end{document}
  

  

• Explicitly specifying the row skip using \\[<len>] where <len> is any length:

  \documentclass{article}
  \begin{document}
  \begin{tabular}{c c}
    $f^{(n)}(x)$ & $f^{(n)}(0)$ \\[1cm] \hline
    $-2xe^{-x^{x^{x^2}}}$ & 0
  \end{tabular}
  \end{document}
  

  

• Modifying the array package's \extrarowheight length using \setlength{\extrarowheight}{<len>}, where <len> is any length:

  \documentclass{article}
  \usepackage{array}% http://ctan.org/pkg/array
  \begin{document}
  \setlength{\extrarowheight}{20pt}
  \begin{tabular}{c c}
    $f^{(n)}(x)$ & $f^{(n)}(0)$ \\\hline
    $-2xe^{-x^{x^{x^2}}}$ & 0
  \end{tabular}
  \end{document}
  

  

In the above examples, the \hline was used to illustrate the effect of the different styles used. The choice depends on the specific usage/typesetting of the tabular within your document.

Finally, if the contents of your entire tabular is math, you could typeset it in an array environment:

\[
  \begin{array}{c c}
    f^{(n)}(x) & f^{(n)}(0) \\
    -2xe^{-x^{x^{x^2}}} & 0
  \end{array}
\]

You can use booktabs. Left is the output using \midrule, right the one with the \hline approach.

The left table is also improved by stating \displaystyle that uses less cramped superscripts.

\documentclass{article}
\usepackage{array,booktabs}
\newcolumntype{C}{>{$\displaystyle}c<{$}}

\begin{document}

\begin{tabular}{CC}
f^{(n)}(x) & f^{(n)}(0) \\
\midrule
-2xe^{-x^{x^{x^2}}} & 0
\end{tabular}
\qquad
\begin{tabular}{cc}
$f^{(n)}(x)$ & $f^{(n)}(0)$ \\
\hline
$-2xe^{-x^{x^{x^2}}}$ & 0
\end{tabular}

\end{document}

You have another option here - specifying row colors:

\documentclass{article}
\usepackage{color, colortbl}
%Light gray:
\definecolor{Gray}{gray}{0.9}

%Light cyan:
\definecolor{LightCyan}{rgb}{0.88,1,1}

\begin{document}
\renewcommand{\arraystretch}{2}
\begin{tabular}{c c}
\rowcolor{Gray}
$f^{(n)}(x)$ & $f^{(n)}(0)$ \\\hline
\rowcolor{LightCyan}
$-2xe^{-x^{x^{x^2}}}$ & 0
\end{tabular}
\end{document}