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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{Differentiation Rules}

\begin{example}
Find the derivative of \[
f(x)=\frac{(x+2)(x^{2}+1)}{(x+3)(x^{2}-2)}.\]

\end{example}

\subsubsection*{Solution}

This is more complicated, since you have products in the numerator and denominator,
as well as the quotient. But the thing to look at is that the last thing you
do in constructing this function is to take the quotient, so the first thing
you need to handle in taking the derivative is the quotient. You always work
from the outside in. 

The first step, then, is to apply the quotient rule:\begin{eqnarray*}
f'(x) & = & \left( \frac{(x+2)(x^{2}+1)}{(x+3)(x^{2}-2)}\right) '\\
 & = & \frac{\left( (x+2)(x^{2}+1)\right) '(x+3)(x^{2}-2)-\left( (x+3)(x^{2}-2)\right) '(x+2)(x^{2}+1)}{(x+3)(x^{2}-2)}.
\end{eqnarray*}



\subsubsection*{Next step}
\end{document}
