\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