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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 \[
\left( \left( x^{2}+3x-2\right) \left( 3x^{4}+x^{2}-1\right) \right) '.\]

\end{example}

\subsubsection*{Solution}

\begin{eqnarray*}
\left( \left( x^{2}+3x-2\right) \left( 3x^{4}+x^{2}-1\right) \right) ' & = & \left( x^{2}+3x-2\right) '\left( 3x^{4}+x^{2}-1\right) +\left( x^{2}+3x-2\right) \left( 3x^{4}+x^{2}-1\right) '\\
 & = & \left( 2x+3\right) \left( 3x^{4}+x^{2}-1\right) +\left( x^{2}+3x-2\right) \left( 12x^{3}+2x\right) ,
\end{eqnarray*}
just using the product rule and the method for differentiating polynomials described
above. 

Now, there is a question as to when you should stop. Do you have to simplify?
Usually, I would say no, don't simplify unless there is some reason to do so.
If you are going to be using this result in some way, to compute something else,
then it's a good idea to simplify so that that computation would become easier.
But if all you have to do is differentiate, leave it at that. The only effect
you can achieve, say on your exam, by simplifying beyond this point is to make
some minor error, which will cause you to lose points. 

If you feel you have to simplify this, then:

\begin{eqnarray*}
\left( \left( x^{2}+3x-2\right) \left( 3x^{4}+x^{2}-1\right) \right) ' & = & \left( x^{2}+3x-2\right) '\left( 3x^{4}+x^{2}-1\right) +\left( x^{2}+3x-2\right) \left( 3x^{4}+x^{2}-1\right) '\\
 & = & \left( 2x+3\right) \left( 3x^{4}+x^{2}-1\right) +\left( x^{2}+3x-2\right) \left( 12x^{3}+2x\right) \\
 & = & (6x^{5}+9x^{4}+2x^{3}+3x^{2}-2x-3)+(12x^{5}+36x^{4}-22x^{3}+6x^{2}-4x)\\
 & = & 18x^{5}+45x^{4}-20x^{3}+9x^{2}-6x-3.
\end{eqnarray*}
Whew. It occurs to me that it would be instructive to have made a mistake in
that calculation. Did I? The point is, that unless you have to use this for
some other purpose, that messy calculation can only do you harm. It also is
easier for me to see what you did, if you would stop as I did above when the
differentiating is all done. 
\end{document}
