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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 the derivative of the function \( f(x)=2^{x} \).
\end{example}

\subsubsection*{Solution}

At any point \( x \), \begin{eqnarray*}
f'(x) & = & \lim _{h\rightarrow 0}\frac{2^{(x+h)}-2^{x}}{h},\, \textrm{by}\, \textrm{the}\, \textrm{definition},\\
 & = & \lim _{h\rightarrow 0}\frac{2^{x}2^{h}-2^{x}}{h},\, \textrm{since}\, 2^{(x+h)}=2^{x}2^{h},\\
 & = & \lim _{h\rightarrow 0}2^{x}\left( \frac{2^{h}-1}{h}\right) ,\, \textrm{factoring}\, \textrm{out}\, 2^{x},\\
 & = & 2^{x}\lim _{h\rightarrow 0}\left( \frac{2^{h}-1}{h}\right) ,
\end{eqnarray*}
so, in order to know this slope, all we need to do is find that last limit,
which is the slope of the curve \[
f(x)=2^{x}\]
 at \( x=0 \). If we're careful in computing it, we'll see that the slope is
about \( 0.69 \).
\end{document}
