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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{thm}
\textbf{\emph{{[}The Power Rule{]}.}} For each positive integer \( n \), \( (x^{n})'=nx^{n-1}. \) 
\end{thm}
\begin{proof}
This works similarly to the specific cases we went over earlier. \textbf{\emph{Audio}}

For \( f(x)=x^{n} \), \begin{eqnarray*}
f'(x) & = & \lim _{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{(x+h)^{n}-(x)^{n}}{h}\\
 & = & \lim _{h\rightarrow 0}\frac{x^{n}+n\, x^{n-1}h+\dots +h^{n}-x^{n}}{h}\, Hint\\
 & = & \lim _{h\rightarrow 0}n\, x^{n-1}+\dots +h^{n-1},\, \textrm{by}\, \textrm{cancelling}\, \textrm{the}\, x^{n}'s\, \textrm{and}\, \textrm{dividing}\, \textrm{by}\, h.\textrm{ }\\
 & = & n\, x^{n-1}.
\end{eqnarray*}
\end{proof}

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