%% 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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.
\setcounter{section}{3}
\setcounter{subsection}{3}
\setcounter{subsubsection}{1}
\makeatother

\begin{document}


\title{On-line Math 21}


\email{david.johnson@lehigh.edu}


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

\maketitle

\subsubsection{Derivatives of \protect\( \sinh (x)\protect \) and \protect\( \cosh (x)\protect \).}

We don't have to use geometric methods to find derivatives of hyperbolic functions;
since 
\[
\sinh (x)\, :=\frac{1}{2}(e^{x}-e^{-x}),\]
it follows directly that 
\[
\sinh '(x)=\frac{1}{2}(e^{x}+e^{-x})=\cosh (x),\]
just as for trig functions. The next rule is a little different, though. Just
like with the rules of hyperbolic trigonometry, there are some sign changes
between trigonometric and hyperbolic rules:
\[
\cosh '(x)=\sinh (x).\]
 

Now, the rest is like real trigonometry: \( \tanh (x):=\sinh (x)/\cosh (x) \),
\( \textrm{sech}(x):=1/\cosh (x) \), \textit{etc.}. 

There are formulas similar to trig functions that do hold: 
\begin{eqnarray*}
\cosh ^{2}x-\sinh ^{2}x & = & 1,\\
(\sinh x)' & = & \cosh x,\\
(\cosh x)' & = & \sinh x,\\
(\tanh x)' & = & \mathrm{sech}^{2}x\\
(\textrm{sech}x)' & = & -\textrm{sech}x\, \tanh x.
\end{eqnarray*}
In fact, all the rules for derivatives of trigonometric functions have mirror-images
as rules for derivatives of hyperbolic functions. The only differences, aside
from the ``h'' in the names, is the occasional sign change, like with \( \cosh '(x)=+\sinh x \).

\end{document}
