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\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{The definition of the derivative}

\begin{example}
Find \( f'(2) \) for the function \( f(x)=x^{3} \). 
\end{example}

\subsubsection*{Solution }

This is similar to the example of the slope of a curve .

\begin{eqnarray*}
f'(2) & = & \lim _{h\rightarrow 0}\frac{f(2+h)-f(2)}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{(2+h)^{3}-(2)^{3}}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{2^{3}+3\cdot 2^{2}h+3\cdot 2h^{2}+h^{3}-2^{3}}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{12h+6h^{2}+h^{3}}{h},
\end{eqnarray*}
cancelling the \( 2^{3} \)'s.
\end{document}
