There are several possible solutions here (discussed in the comments).
The first is
\documentclass{article}
\usepackage{booktabs}
\usepackage[margin=13mm,paper=a4paper]{geometry}
\usepackage{graphics}
\begin{document}
\begin{table}
\centering
\small
\begin{tabular*}{\linewidth}{@{\extracolsep{\fill}}l c}
\toprule
y & y' \\
\midrule
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
\bottomrule
\end{tabular*}
\end{table}
\end{document}
which results in
The second is
\documentclass{article}
\usepackage{booktabs}
\usepackage[margin=13mm,paper=a4paper]{geometry}
\usepackage{graphics}
\begin{document}
\begin{table}
\centering
\small
\begin{tabular}{l c}
\toprule
y & y' \\
\midrule
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
\bottomrule
\end{tabular}
\end{table}
\end{document}
which results in
The third is
\documentclass{article}
\usepackage{booktabs}
\usepackage[margin=13mm,paper=a4paper]{geometry}
\usepackage{graphics}
\begin{document}
\begin{table}
\centering
\small
\resizebox{!}{391pt}{
\begin{tabular}{l c}
\toprule
y & y' \\
\midrule
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
\bottomrule
\end{tabular}}
\end{table}
\end{document}
which results in
(It's hard to tell from the picture, but it fills up the whole page and doesn't overflow.)
The fourth is
\documentclass{article}
\usepackage{booktabs}
\usepackage[margin=13mm,paper=a4paper]{geometry}
\usepackage{graphics}
\usepackage{tabularx}
% ...
\begin{document}
\begin{table}
\centering
\small
\begin{tabularx}{\textwidth}{ XXXXXXX }
\toprule
y & y' \\
\midrule
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
as & well \\
using & the \\
\bottomrule
\end{tabularx}
\end{table}
\end{document}
Which gives
I may be incorrectly spacing the tabularx option, but even still, I personally think the third option is the best option.
Thanks to all the commenters (Fran, David Carlisle, Johannes_B, and Heiko Oberdiek) for their suggestions.
Finally, in light of your comments, I'm working on a solution to what you really want to create. So far, I have
\documentclass{article}
\usepackage{booktabs}
\usepackage[margin=13mm,paper=a4paper]{geometry}
\usepackage{graphics}
% ...
\begin{document}
\begin{table}
\centering
\small
\resizebox{520pt}{320pt}{%
\begin{tabular}{l l}
Derivatives and Integrals \\
\\
\toprule
Basic Differentiation Rules \\
\midrule
\\
1. $\frac {d}{dx} [cu] = cu'$ & 2. $\frac{d}{dx} [u \pm v] = u' \pm
v'$ \\
3. $\frac{d}{dx} [uv] = uv' + vu'$ & 4. $\frac{d}{dx} [\frac{u}{v}]
= \frac{vu' - uv'}{v^2}$ \\
5. $\frac{d}{dx} [c] = 0$ & 6. $\frac{d}{dx} [u^n] = nu^{n-1} \quad
u'$ \\
7. $\frac{d}{dx} [x] = 1$ & 8. $\frac{d}{dx} [\mid u \mid] =
\frac{u}{\mid u \mid} (u'), \quad u \neq 0$ \\
9. $\frac{d}{dx} [ln \quad u] = \frac{u'}{u}$ & 10. $\frac{d}{dx}
[e^u] = e^u \quad u'$ \\
11. $\frac{d}{dx} [sin \quad u] = (cos \quad u) u'$ & 12. $\frac{d}
{dx} [cos \quad u] = -(sin \quad u) u'$ \\
13. $\frac{d}{dx} [tan \quad u] = (sec^2 \quad u)u'$ & 14.
$\frac{d}{dx} [cot \quad u] = -(csc^2 \quad u) u'$ \\
15. $\frac{d}{dx} [sec \quad u] = (sec \quad u \quad tan \quad u)
u'$ & 16. $\frac{d}{dx} [csc \quad u] = -(csc \quad u \quad cot
\quad u) u'$ \\
17. $\frac{d}{dx} [arcsin \quad u] = \frac{u'}{\sqrt{-1 - u^2}}$ &
18. $\frac{d}{dx} [arccos \quad u] = \frac{-u'}{\sqrt{1-u^2}}$ \\
19. $\frac{d}{dx} [arctan \quad u] = \frac{u'}{1 + u^2}$ & 20.
$\frac{d}{dx} [arccot \quad u] = \frac{-u'}{1 + u^2}$ \\
21. $\frac{d}{dx} [arcsec \quad u] = \frac{u'}{\mid u \mid
\sqrt{u^2 - 1}}$ & 22. $\frac{d}{dx} [arcsec \quad u] = \frac{-u'}
{\mid u \mid \sqrt{u^2 - 1}}$ \\
\\
Basic Integration Formulas \\
\\
1. $\int k \quad f(u) \quad d u = k \int f(u) \quad du$ & 2. $\int
\quad [f(u) \pm g (u)] \quad du = \int f(u) \quad du \pm \int g(u)
\quad du$ \\
3. $\int d u = u + C$ & 4. $\int u^n d u = \frac{u^{n+1}}{n + 1} +
C, \quad n \neq -1$ \\
5. $\int \frac{d}{u} = ln \mid u \mid + C$ & 6. $\int e^u d u = e^u
+ C$ \\
7. $\int sin \quad u \quad du = -cos u + C$ & 8. $\int cos \quad u
\quad d u = sin \quad u + C$ \\
9. $\int tan \quad u \quad du = -ln \mid cos \quad u \mid + C$ &
10. $\int cot \quad u \quad du = ln \mid sin \quad u \mid + C$ \\
11. $\int sec \quad u \quad du = ln \mid sec \quad u + tan \quad du
\mid + C$ & 12. $\int csc \quad u \quad du = -ln \mid csc \quad u +
cot \quad u \mid + C$ \\
13. $\int sec^2 u \quad du = tan \quad u + C$ & 14. $\int csc^2
\quad u \quad du = -cot \quad u + C$ \\
15. $\int sec \quad u \quad tan \quad u \quad du = sec \quad u + C$
& 16. $\int csc \quad u \quad cot \quad du = -csc \quad u + C$ \\
17. $\int \frac{du}{\sqrt{a^2 - u^2}} = arcsin \frac{u}{a} + C$ &
18. $\int \frac{du}{a^2 + u^2} = \frac{1}{a} arctan \frac{u}{a} +
C$ \\
19. $\int \frac{du}{u \sqrt{u^2 - a^2}} = \frac{1}{a} arcsec
\frac{\mid u \mid}{a} + C$ \\
\bottomrule
\end{tabular}}
\end{table}
\end{document}
Which gives
I am working toward this:
Some of the lines aren't centered, and the resizing isn't perfect, but you can fiddle with the resizing. I'll try to figure out how to center the lines properly.
Hope this helps!
\columnwidth. What's the purpose of the\resizebox?tabularxto get a full page width table.\resizebox{\columnwidth}{!}{%on tables, you could of course make it full width by putting lots of space between the columns but that just makes it harder to read, why not let the table be natural width but centred on the page.