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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 }

The next step is to multiply top and bottom by \( (\sqrt{x+h}+\sqrt{x}) \).

\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) }.
\end{eqnarray*}
This trick is known as \emph{conjugation}. Any time we have a difference of
square roots, we can simplify that by multiplying, numerator and denomonator,
by the sum of the square roots (the \emph{conjugate}). The idea, contrary to
what you were probably taught before, is to rationalize the \textbf{numerator},
\emph{not} the denominator. We can do more with the numerator algebraically
once we rationalize it than we could with the denominator. I never have understood
why you are supposed to rationalize the denominator in fractions. I think that
\[
\frac{1}{\sqrt{2}}\]
makes more sense than \[
\frac{\sqrt{2}}{2},\]
anyway. 

Now that we have multiplied by this conjugate, what happens?


\subsubsection*{Next}
\end{document}
