%% LyX 1.1 created this file.  For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass[11pt,oneside]{amsart}
\usepackage[T1]{fontenc}
\usepackage{geometry}
\geometry{verbose,letterpaper,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\setlength\parskip{\medskipamount}
\setlength\parindent{0pt}

\makeatletter


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.
\providecommand{\LyX}{L\kern-.1667em\lower.25em\hbox{Y}\kern-.125emX\@}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.
 \theoremstyle{plain}    
 \newtheorem{thm}{Theorem}[section]
 \numberwithin{equation}{section} %% Comment out for sequentially-numbered
 \numberwithin{figure}{section} %% Comment out for sequentially-numbered
 \theoremstyle{plain}    
 \newtheorem*{thm*}{Theorem} 
 \theoremstyle{definition}
  \newtheorem{example}[thm]{Example}
 \theoremstyle{definition}
  \newtheorem{xca}[section]{Exercise}%%Delete [section] for sequential numbering

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{3}
\setcounter{subsection}{0}
\setcounter{subsubsection}{2}

\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


\urladdr{http://www.lehigh.edu/\textasciitilde{}dlj0}

\maketitle

\subsection{Trigonometric functions}


\subsubsection{Derivatives of \protect\( \sin (x)\protect \) and \protect\( \cos (x)\protect \).}


\subsection*{The derivative of \protect\( \sin x\protect \).}

The derivative of the sine function \( \sin (x) \) (or \( \sin x \)) is the
first derivative you can't figure out from the definition just using enough
algebra to cancel a power of \( h \) top and bottom. However, what it does
take is the trigonometric limits we just derived.

\begin{thm*}
The functions \( \sin x \) and \( \cos x \) have the following derivatives:
\begin{eqnarray*}
(\sin x)' & = & \cos x\\
(\cos x)' & = & -\sin x
\end{eqnarray*}

\end{thm*}
\begin{proof}
~
\end{proof}
\begin{example}
Find \( (\tan x)' \). 
\end{example}

\subsubsection*{Solution}

\begin{example}
Find \( (\sec x)' \). 
\end{example}

\subsubsection*{Solution}

\begin{xca}
Find \( (\cot x)' \). 
\end{xca}
This is similar to the previous ones. Watch for signs (\( \pm  \)).

\begin{xca}
Find \( (\csc x)' \). 
\end{xca}
\begin{example}
Find \( (x^{3}\cos x)' \). 
\end{example}

\subsubsection*{Solution}

\begin{xca}
Find \( (\sin x\cos x)' \). 
\end{xca}
\begin{example}
Find the tangent line to the curve \( y=\sin (x) \) at \( (\pi /6,1/2) \),
and use that to find, approximately, \( \sin (\pi /6+0.1) \). 
\end{example}

\subsubsection*{Solution }
\end{document}
