\magnification=1100
\overfullrule0pt

\input amssym.def
\input prepictex
\input pictex
\input postpictex


% ********************* Definitions ************************************

%\def\widetilde{\mathaccent"0365 }

\def\CC{{\Bbb C}}
\def\FF{{\Bbb F}}
\def\HH{{\Bbb H}}
\def\NN{{\Bbb N}}
\def\OO{{\Bbb O}}
\def\QQ{{\Bbb Q}}
\def\RR{{\Bbb R}}
\def\ZZ{{\Bbb Z}}

\def\cA{{\cal A}}

\def\cB{{\cal B}}
\def\cR{{\cal R}}
\def\cZ{{\cal Z}}
\def\cC{{\cal C}}
\def\cR{{\cal R}}

\def\fb{\frak{b}}
\def\fg{\frak{g}}
\def\fh{\frak{h}}
\def\fn{\frak{n}}

\def\Card{\hbox{Card}}
\def\End{\hbox{End}}
\def\Hom{\hbox{Hom}}
\def\Ind{\hbox{Ind}}
\def\Id{\hbox{Id}}
\def\codim{\hbox{codim}}

\def\diag{\hbox{diag}}
\def\id{\hbox{id}}
\def\im{\hbox{im}}
\def\tr{\hbox{tr}}
\def\Tr{\hbox{Tr}}

\def\sech{\hbox{sech}}
\def\csch{\hbox{csch}}
\def\coth{\hbox{coth}}

\def\fbox{}


% ********************* FONTS ************************************

\font\smallcaps=cmcsc10
\font\titlefont=cmr10 scaled \magstep1
\font\titlefontbold=cmbx10 scaled \magstep1
\font\titlesubfont=cmr10 scaled \magstep1
\font\sectionfont=cmbx10
\font\tinyrm=cmr10 at 8pt

% ******************** SECTION HEADERS ***************************

\newcount\sectno
\newcount\subsectno
\newcount\resultno

\def\section #1. #2\par{
\sectno=#1
\resultno=0
\bigskip\noindent{\sectionfont #1.  #2}~\medbreak}

\def\subsection #1\par{\bigskip\noindent{\it  #1} \medbreak}

%******************* MATHEMATICAL LABELS **************************

\def\prop{ \global\advance\resultno by 1
\medskip\noindent{\bf Proposition \the\sectno.\the\resultno. }\sl}
\def\lemma{ \global\advance\resultno by 1
\medskip\noindent{\bf Lemma \the\sectno.\the\resultno. }
\sl}
\def\fact{ \global\advance\resultno by 1
\medskip\noindent{\bf Fact \the\sectno.\the\resultno. }
\sl}

\def\remark{ \global\advance\resultno by 1
\medskip\noindent{\bf Remark \the\sectno.\the\resultno. }}
\def\example{ \global\advance\resultno by 1
\medskip\noindent{\bf Example \the\sectno.\the\resultno. }\sl}
\def\cor{ \global\advance\resultno by 1
\medskip\noindent{\bf Corollary \the\sectno.\the\resultno. }\sl}
\def\thm{ \global\advance\resultno by 1
\medskip\noindent{\bf Theorem \the\sectno.\the\resultno. }\sl}
\def\defn{ \global\advance\resultno by 1
\medskip\noindent{\it Definition \the\sectno.\the\resultno. }\slrm}
\def\endthm{\rm\medskip}
\def\thmend{\rm\medskip}
\def\endlemma{\rm\medskip}
\def\endfact{\rm\medskip}
\def\endexample{\rm\medskip}
\def\endprop{\rm\medskip}
\def\endcor{\rm\medskip}
\def\pf{\rm\smallskip\noindent{\it Proof. }}
\def\endpf{\qed\hfil\medskip}
\def\pfend{\qed\hfil\medskip}
\def\note{\smallbreak\noindent{Note:}}
\def\enddefn{\rm\medskip}

%Homemade Struts:
\newbox\strutAbox
\setbox\strutAbox=\hbox{\vrule height 12pt depth6pt width0pt}
\def\strutA{\relax\copy\strutAbox}
\newbox\strutBbox
\setbox\strutBbox=\hbox{\vrule height 10pt depth5pt width0pt}
\def\strutB{\relax\copy\strutBbox}
\newbox\strutDbox
\setbox\strutDbox=\hbox{\vrule height 11pt depth5pt width0pt}
\def\strutD{\relax\copy\strutDbox}
%high strut:
\newbox\strutHbox
\setbox\strutHbox=\hbox{\vrule height 11pt depth1pt width0pt}
\def\strutH{\relax\copy\strutHbox}
%low strut:
\newbox\strutLbox
\setbox\strutLbox=\hbox{\vrule height 1pt depth5pt width0pt}
\def\strutL{\relax\copy\strutLbox}



% hack to ignore lots of typed stuff....
\def\ignore#1{\relax}

% ******************  QED SIGNS  *********************************

\def\qed{\hbox{\hskip 1pt\vrule width4pt height 6pt depth1.5pt \hskip 1pt}}

\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt
\vrule width.2pt}
\hrule height.2pt}}}}
\def\square{\mathchoice\sqr54\sqr54\sqr{3.5}3\sqr{2.5}3}
\def\whiteslug{\bf $ \square $ \rm}  % open square


%*************** EQUATIONS WITH NUMBERS **************

\def\formula{\global\advance\resultno by 1
\eqno{(\the\sectno.\the\resultno)}}
\def\formulano{\global\advance\resultno by 1 (\the\sectno.\the\resultno)}
\def\tableno{\global\advance\resultno by 1
\the\sectno.\the\resultno. }
\def\lformula{\global\advance\resultno by 1
\leqno(\the\sectno.\the\resultno)}

%************Commutative diagrams**********************

\def\mapright#1{\smash{\mathop
        {\longrightarrow}\limits^{#1}}}

\def\mapleftright#1{\smash{\mathop
        {\longleftrightarrow}\limits^{#1}}}


\def\mapsrightto#1{\smash{\mathop
        {\longmapsto}\limits^{#1}}}

\def\mapleft#1{\smash{
   \mathop{\longleftarrow}\limits^{#1}}}

\def\mapdown#1{\Big\downarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}

\def\lmapdown#1{{\hbox{$\scriptstyle#1$}}
\llap {$\vcenter{\hbox{\Big\downarrow}}$} }

\def\mapup#1{\Big\uparrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapne#1{\Big\nearrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapse#1{
%{$\vcenter{
\hbox{$\scriptstyle#1$}
%$}
\rlap{ $\vcenter{\hbox{$\searrow$}}$ }  }
\def\mapnw#1{\Big\nwarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapsw#1{
%\Big
\swarrow
   \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}


%********** DATING ******************************************
\def\monthname {\ifcase\month\or January\or February\or March\or April\or
May\or June\or
July\or August\or September\or October\or November\or December\fi}

\newcount\mins  \newcount\hours  \hours=\time \mins=\time
\def\now{\divide\hours by60 \multiply\hours by60 \advance\mins by-\hours
     \divide\hours by60         % NOTE: \divide only gives integer answers.
     \ifnum\hours>12 \advance\hours by-12
       \number\hours:\ifnum\mins<10 0\fi\number\mins\ P.M.\else
       \number\hours:\ifnum\mins<10 0\fi\number\mins\ A.M.\fi}
\def\today {\monthname\ \number\day, \number\year}


%**************** PAGE HEADERS *************************

\nopagenumbers
\def\runningtitle{\smallcaps trig derivatives}
\headline={\ifnum\pageno>1\eoheadline\else\firstheadline\fi}
\def\names{\smallcaps a.\ ram}
%\def\firstheadline{\noindent Preliminary Draft \hfill  \today}
\def\firstheadline{}
\def\eoheadline{\ifodd\pageno\oddheadline\else\evenheadline\fi}
\def\oddheadline{\tenrm\hfil\runningtitle\hfil\folio}
\def\evenheadline{\tenrm\folio\hfil{\names}\hfil}


%**************** TITLE *************************
\vphantom{$ $}  %My kludge to get the first page to move down a bit
\vskip.75truein
\centerline{\titlefont Derivatives of trig functions}
\bigskip
\centerline{\rm Arun Ram}
%${}^\ast$ 
\centerline{Department of Mathematics}
\centerline{University of Wisconsin, Madison}
\centerline{Madison, WI 53706 USA}
\centerline{{\tt ram@math.wisc.edu}}
\medskip
\centerline{Version: \today}

%\footnote{}{\tinyrm 
%${}^\ast$ 
%Research partially supported by the National Security Agency
%and by EPSRC Grant GR K99015 at the Newton Institute for
%Mathematical Sciences.}
%\footnote{}{\tinyrm
%\noindent {Keywords:} exponential functions, trigonometric functions}

\bigskip

%**************** ABSTRACT *************************
%\noindent{\bf Abstract.}


\bigskip\bigskip
Define
$$\eqalign{
e^x &= 1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots, \cr
\sin x &= x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}
+{x^9\over 9!}-{x^{11}\over 11!}+{x^{13}\over 13!} -\cdots, \cr
\cos x &= 1-{x^2\over 2!}+{x^4\over 4!}-{x^6\over 6!}
+{x^8\over 8!}-{x^{10}\over 10!}+{x^{12}\over 12!} -\cdots, \cr
}
$$
and
$$\tan x ={\sin x\over \cos x}, \quad
\cot x ={1\over \tan x}, \quad
\sec x ={1\over \cos x}, \quad
\csc x ={1\over \sin x}. 
$$


\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d e^x\over dx} }$.
\bigskip
$$\eqalign{
{d e^x\over dx} 
&= {d\ \over dx}\big(1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots\big) \cr
&= 0+1+{1\over 2!}2x+{1\over 3!}3x^2+{1\over 4!}4x^3
+{1\over 5!}5x^4+{1\over 6!}6x^5+{1\over 7!}7x^6 +\cdots \cr
&= 1+{1\over 2}2x+{1\over 3\cdot 2!}3x^2+{1\over 4\cdot 3!}4x^3
+{1\over 5\cdot 4!}5x^4+{1\over 6\cdot 5!}6x^5+{1\over 7\cdot 6!}7x^6 
+\cdots \cr
&= 1+x+{x^2\over 2!}+{x^3\over 3!}+{x^4\over 4!}
+{x^5\over 5!}+{x^6\over 6!}+{x^7\over 7!} +\cdots \cr
&=e^x. \cr}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d \sin x\over dx} }$.
\bigskip
$$\eqalign{
{d \sin x\over dx} 
&= {d\ \over dx}
\big(x-{x^3\over 3!}+{x^5\over 5!}-{x^7\over 7!}
+{x^9\over 9!}-{x^{11}\over 11!}+{x^{13}\over 13!} -\cdots\big) \cr
&= 1-{1\over 3!}3x^2+{1\over 5!}5x^4-{1\over 7!}7x^6
+{1\over 9!}9x^8-{1\over 11!}11x^{10}+{1\over 13!}13x^{12} -\cdots \cr
&= 1-{1\over 2!}x^2+{1\over 4!}x^4-{1\over 6!}x^6
+{1\over 8!}x^8-{1\over 10!}x^{10}+{1\over 12!}x^{12} -\cdots \cr
&=\cos x. \cr}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d \cos x\over dx} }$.
\bigskip
$$\eqalign{
{d \cos x\over dx} 
&= {d\ \over dx}
\big(1-{x^2\over 2!}+{x^4\over 4!}-{x^6\over 6!}
+{x^8\over 8!}-{x^{10}\over 10!}+{x^{12}\over 12!} -\cdots\big) \cr
&=0-{1\over 2!}2x+{1\over 4!}4x^3-{1\over 6!}6x^5
+{1\over 8!}8x^7-{1\over 10!}10x^9+{1\over 12!}12x^11 -\cdots \cr
&=-x+{1\over 3!}x^3-{1\over 5!}x^5
+{1\over 7!}x^7-{1\over 9!}x^9+{1\over 11!}1x^11 -\cdots \cr
&=-\big(x-{1\over 3!}x^3+{1\over 5!}x^5
-{1\over 7!}x^7+{1\over 9!}x^9-{1\over 11!}1x^11 +\cdots\big) \cr
&= - \sin x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d \tan x\over dx} }$.
\bigskip
$$\eqalign{
{d \tan x\over dx} 
&= {d\ \over dx}
\left({\sin x\over \cos x}\right)
= {d\ \over dx}
\left(\sin x(\cos x)^{-1}\right) \cr
&= \sin x {d(\cos x)^{-1}\over dx}
+{d\sin x\over dx}(\cos x)^{-1} \cr
&= \sin x (-1)(\cos x)^{-2}{d\cos x\over dx}
+\cos x\,\cdot{1\over \cos x} \cr
&= -{\sin x\over \cos^2 x}(-\sin x)+1 
= {\sin^2 x\over \cos^2 x}+1 \cr
&= {\sin^2 x+\cos^2 x\over \cos^2 x} 
= {1\over \cos^2 x} 
= {\sec^2 x}. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d \sec x\over dx} }$.
\bigskip
$$\eqalign{
{d \sec x\over dx} 
&= {d\ \over dx}
\left({1\over \cos x}\right)
= {d\ \over dx}
\left((\cos x)^{-1}\right) 
= (-1)(\cos x)^{-2}{d\cos x\over dx} \cr
&= -{1\over \cos^2 x}(-\sin x) 
= {\sin x\over \cos^2 x} 
= {\sin x\over \cos x}\cdot{1\over \cos x} 
=\tan x\sec x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d \csc x\over dx} }$.
\bigskip
$$\eqalign{
{d \csc x\over dx} 
&= {d\ \over dx}
\left({1\over \sin x}\right)
= {d\ \over dx}
\left((\sin x)^{-1}\right) 
= (-1)(\sin x)^{-2}{d\sin x\over dx} \cr
&= -{1\over \sin^2 x}(\cos x) 
= -{\cos x\over \sin^2 x} 
= -{\cos x\over \sin x}\cdot{1\over \sin x} 
=-\cot x\csc x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d \cot x\over dx} }$.
\bigskip
$$\eqalign{
{d \cot x\over dx} 
&= {d\ \over dx}
\left({\cos x\over \sin x}\right)
= {d\ \over dx}
\left(\cos x(\sin x)^{-1}\right) \cr
&= \cos x {d(\sin x)^{-1}\over dx}
+{d\cos x\over dx}(\sin x)^{-1} \cr
&= \cos x (-1)(\sin x)^{-2}{d\sin x\over dx}
+\,-(\sin x)\,\cdot{1\over \sin x} \cr
&= -{\cos x\over \sin^2 x}\cdot\cos x-1 
= {-\cos^2 x\over \sin^2 x}-1 \cr
&= {-\cos^2 x-\sin^2 x\over \sin^2 x} 
= {-1\over \sin^2 x} 
= {\csc^2 x}. \cr
}$$


\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d\ \sinh x\over dx} }$.
\bigskip
$$\eqalign{
{d\ \sinh x\over dx} 
&= {d\ \over dx}
\big(x+{x^3\over 3!}+{x^5\over 5!}+{x^7\over 7!}
+{x^9\over 9!}+{x^{11}\over 11!}+{x^{13}\over 13!} +\cdots\big) \cr
&= 1+{1\over 3!}3x^2+{1\over 5!}5x^4+{1\over 7!}7x^6
+{1\over 9!}9x^8+{1\over 11!}11x^{10}+{1\over 13!}13x^{12} +\cdots \cr
&= 1+{1\over 2!}x^2+{1\over 4!}x^4+{1\over 6!}x^6
+{1\over 8!}x^8+{1\over 10!}x^{10}+{1\over 12!}x^{12} +\cdots \cr
&=\cosh x. \cr}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d\ \cosh x\over dx} }$.
\bigskip
$$\eqalign{
{d\ \cosh x\over dx} 
&= {d\ \over dx}
\big(1+{x^2\over 2!}+{x^4\over 4!}+{x^6\over 6!}
+{x^8\over 8!}+{x^{10}\over 10!}+{x^{12}\over 12!} +\cdots\big) \cr
&=0+{1\over 2!}2x+{1\over 4!}4x^3+{1\over 6!}6x^5
+{1\over 8!}8x^7+{1\over 10!}10x^9+{1\over 12!}12x^11 +\cdots \cr
&=x+{1\over 3!}x^3+{1\over 5!}x^5
+{1\over 7!}x^7+{1\over 9!}x^9+{1\over 11!}1x^11 +\cdots \cr
&= \sinh x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d\ \tanh x\over dx} }$.
\bigskip
$$\eqalign{
{d\ \tanh x\over dx} 
&= {d\ \over dx}
\left({\sinh x\over \cosh x}\right)
= {d\ \over dx}
\left(\sinh x(\cosh x)^{-1}\right) \cr
&= \sinh x {d(\cosh x)^{-1}\over dx}
+{d\sinh x\over dx}(\cosh x)^{-1} \cr
&= \sinh x (-1)(\cosh x)^{-2}{d\cosh x\over dx}
+\cosh x\,\cdot{1\over \cosh x} \cr
&= -{\sinh x\over \cosh^2 x}\cdot\sinh x+1 
= -{\sinh^2 x\over \cosh^2 x}+1 \cr
&= {-\sinh^2 x+\cosh^2 x\over \cosh^2 x} 
= {1\over \cosh^2 x} 
= {\sech^2 x}. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d\ \sech x\over dx} }$.
\bigskip
$$\eqalign{
{d\ \sech x\over dx} 
&= {d\ \over dx}
\left({1\over \cosh x}\right)
= {d\ \over dx}
\left((\cosh x)^{-1}\right) 
= (-1)(\cosh x)^{-2}{d\cosh x\over dx} \cr
&= -{1\over \cosh^2 x}\cdot\sinh x 
= -{\sinh x\over \cosh^2 x} 
= -{\sinh x\over \cosh x}\cdot{1\over \cosh x} 
=-\tanh x\,\sech x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d\ \csch x\over dx} }$.
\bigskip
$$\eqalign{
{d\ \csch x\over dx} 
&= {d\ \over dx}
\left({1\over \sinh x}\right)
= {d\ \over dx} \left((\sinh x)^{-1}\right) 
= (-1)(\sinh x)^{-2}{d\sinh x\over dx} \cr
&= -{1\over \sinh^2 x}(\cosh x) 
= -{\cosh x\over \sinh^2 x} 
= -{\cosh x\over \sinh x}\cdot{1\over \sinh x} 
=-\,\coth x\,\csch x. \cr
}$$

\bigskip\noindent
{\bf Example:}  Find $\displaystyle{ {d\ \coth x\over dx} }$.
\bigskip
$$\eqalign{
{d\ \coth x\over dx} 
&= {d\ \over dx}
\left({1\over \tanh x}\right)
= {d (\tanh x)^{-1}\over dx} 
= (-1)(\tanh x)^{-2} {d\tanh x\over dx} \cr
&= -{1\over \tanh^2 x} {d\tanh x\over dx} 
= -{1\over \tanh^2 x}\cdot \sech^2 x \cr
&= -{1\over {\sinh^2 x\over \cosh^2 x}}\cdot 
{1\over \cosh^2 x} 
= -{1\over \sinh^2 x} 
= -{\csch^2 x}. \cr
}$$





\vfill\eject
\end

