Design a multiple columns environment to arrange exercises

The result of your earlier question, possibly in a minipage, and an ordinary enumeration environment should get the result you want.

\documentclass{article}
\usepackage{shortlst}% http://ctan.org/pkg/shortlst
\begin{document}

\newenvironment{exercises}[1]
    {#1:\\ \begin{minipage}[t]{0.9\textwidth}\begin{shortenumerate}}
    {\end{shortenumerate}\end{minipage}}

\begin{enumerate}
\item 
\begin{exercises}{\textbf{Differentiation}\\ Differentiate with respect to $x$}
     \item $x^3-3x$ 
     \item $x^5-\frac{x^3}{3}$
     \item $x+\frac{1}{x}$
     \item $(x+1)^2(x-1)^2$
     \item $x\sin x$
\end{exercises}
\item 
\begin{exercises}{Differentiate with respect to $x$}
     \item $x^3-3x$ 
     \item $x^5-\frac{x^3}{3}$
     \item $x+\frac{1}{x}$
     \item $(x+1)^2(x-1)^2$
     \item $x\sin x$
\end{exercises}
\end{enumerate}

\end{document}

which looks like

If you want a table like environment to handle a specific number of columns of equations, then you can define a new counter to do this.

\newcounter{innerlist} 
\renewcommand{\theinnerlist}{\arabic{innerlist}} 
\newcommand{\inneritem}[1]{\refstepcounter{innerlist}\theinnerlist.\ #1}

\newenvironment{exercises}[2]
    {\setcounter{innerlist}{0}\begin{minipage}[t]{0.9\textwidth}#1:\\ \begin{tabular}{#2}}
    {\end{tabular}\end{minipage}}


\begin{enumerate}
    \item 
    \begin{exercises}{\textbf{Differentiation}\\ Differentiate with respect to $x$}{ll}
        \inneritem{$x^3-3x$} & \inneritem{$x^5-\frac{x^3}{3}$}\\
        \inneritem{$x+\frac{1}{x}$} & \inneritem{$(x+1)^2(x-1)^2$}\\
        \inneritem{$x\sin x$} & \inneritem{$x\sin x$}
    \end{exercises}
    \item
    \begin{exercises}{\textbf{Differentiation}\\ Differentiate with respect to $x$}{lll}
        \inneritem{$x^3-3x$} & \inneritem{$x^5-\frac{x^3}{3}$} & \inneritem{$x+\frac{1}{x}$}\\
        \inneritem{$(x+1)^2(x-1)^2$} & \inneritem{$x\sin x$} & \inneritem{$x\sin x$}
    \end{exercises}
\end{enumerate}

This version shouldn't have any big issues spreading over pages. Each minipage shouldn't spread between pages, but the items in the outer enumerated list will spread fine.

An alteration using the longtables package allows this to run over pages:

\newenvironment{exercises}[2]
    {\setcounter{innerlist}{0}#1:\\ \begin{longtable}[t]{#2}}
    {\end{longtable}}

I would suggest to combine Mark's answer with package scrextend. That would support page breaks inside the exercise:

\documentclass{article}
\usepackage{shortlst}% http://ctan.org/pkg/shortlst
\usepackage{scrextend}
\usepackage{lipsum}
\begin{document}

\lipsum[1-4]
\begin{enumerate}
\item {Differentiate with respect to $x$:
  \begin{addmargin}[0pt]{2em}
    \begin{shortenumerate}
      \item $x^3-3x$ 
      \item $x^5-\frac{x^3}{3}$
      \item $x+\frac{1}{x}$
      \item $(x+1)^2(x-1)^2$
      \item $x\sin x$
    \end{shortenumerate}
  \end{addmargin}
}
\item {Differentiate with respect to $x$:
  \begin{addmargin}[0pt]{2em}
    \begin{shortenumerate}
      \item $y=x\cos \frac{1}{2}x$
      \item $y=\sin\left(\frac{\pi x}{180}\right)$
      \item $y=\frac{x^2}{1+x}$
      \item $y=x\cos \frac{1}{2}x$
      \item $y=\sin\left(\frac{\pi x}{180}\right)$
      \item $y=\frac{x^2}{1+x}$
      \item $y=x\cos \frac{1}{2}x$
      \item $y=\sin\left(\frac{\pi x}{180}\right)$
      \item $y=\frac{x^2}{1+x}$
      \item $y=x\cos \frac{1}{2}x$
      \item $y=\sin\left(\frac{\pi x}{180}\right)$
      \item $y=\frac{x^2}{1+x}$
    \end{shortenumerate}
  \end{addmargin}
}
\end{enumerate}
\end{document}

Note: If you're already using a KOMA-Script class you don't need to load the package scrextend.

You may define an enviroment for this:

\documentclass{article}
\usepackage{shortlst}% http://ctan.org/pkg/shortlst  
\usepackage{scrextend}
\usepackage{lipsum}% 
\usepackage[width=10cm]{geometry} %added

\newenvironment{exercise}[1][\strut]{%
  \item #1\begin{addmargin}[0pt]{2em}\begin{shortenumerate} 
}{%
  \end{shortenumerate}\end{addmargin} 
}
\begin{document} 

\lipsum[1] 
\begin{enumerate}
  \begin{exercise}[{Differentiate with respect to $x$:}]
  \item $x^3-3x$  
  \item $x^5-\frac{x^3}{3}$ 
  \item $x+\frac{1}{x}$ 
  \item $(x+1)^2(x-1)^2$ 
  \item $x\sin x$ 
  \end{exercise}
  \begin{exercise}[{Differentiate with respect to $x$:}]
  \item $y=x\cos \frac{1}{2}x$ 
  \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
  \item $y=\frac{x^2}{1+x}$ 
  \item $y=x\cos \frac{1}{2}x$ 
  \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
  \item $y=\frac{x^2}{1+x}$ 
  \item $y=x\cos \frac{1}{2}x$ 
  \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
  \item $y=\frac{x^2}{1+x}$ 
  \item $y=x\cos \frac{1}{2}x$ 
  \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
  \item $y=\frac{x^2}{1+x}$ 
  \end{exercise} 
\end{enumerate} 
\end{document}

You could use a combination of multicols with a newlist using the enumitem package, as demonstrated below.

It allows pagebreaks too.

\documentclass{article} 
\usepackage{lipsum} 
\usepackage{multicol}             % multicols
\usepackage{enumitem}             % customized lists
\usepackage[width=10cm]{geometry} % page dimensions

% setup a new list (terminology from enumitem)
\newlist{mylist}{enumerate}{5}
\setlist[mylist]{label=(\alph*)}

\newenvironment{exercises}[1]{\begin{multicols}{#1}\begin{mylist}}{\end{mylist}\end{multicols}}

\begin{document} 

\vspace*{4cm}   % purely to demonstrate the pagebreak
\lipsum[1-2] 

\begin{enumerate} 
 \item {Differentiate with respect to $x$:}
  \begin{exercises}{2}
      \item $x^3-3x$  
      \item $x^5-\frac{x^3}{3}$ 
      \item $x+\frac{1}{x}$ 
      \item $(x+1)^2(x-1)^2$ 
      \item $x\sin x$ 
  \end{exercises} 
 \item Differentiate with respect to $x$:
 \begin{exercises}{2}
      \item $y=x\cos \frac{1}{2}x$ 
      \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
      \item $y=\frac{x^2}{1+x}$ 
      \item $y=x\cos \frac{1}{2}x$ 
      \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
      \item $y=\frac{x^2}{1+x}$ 
      \item $y=x\cos \frac{1}{2}x$ 
      \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
      \item $y=\frac{x^2}{1+x}$ 
      \item $y=x\cos \frac{1}{2}x$ 
      \item $y=\sin\left(\frac{\pi x}{180}\right)$ 
      \item $y=\frac{x^2}{1+x}$ 
  \end{exercises} 
\end{enumerate} 
\end{document}