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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 \( f'(x) \) for the function \( f(x)=\sqrt{x} \). 
\end{example}

\subsubsection*{Solution }

\begin{eqnarray*}
f'(x) & = & \lim _{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{\left( \sqrt{x+h}-\sqrt{x}\right) \left( \sqrt{x+h}+\sqrt{x}\right) }{h\left( \sqrt{x+h}+\sqrt{x}\right) }\\
 & = & \lim _{h\rightarrow 0}\frac{(x+h)-x}{h(\sqrt{x+h}+\sqrt{x})},
\end{eqnarray*}
multiplying out the numerator. Remember, \[
(a-b)(a+b)=a^{2}-b^{2}.\]
Note, now, that the numerator is just \( h \). Cancel that out, top and bottom,
which will mean that the limit is no longer an indeterminate form of type \( 0/0 \). 


\subsubsection*{Finish it}
\end{document}
