\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\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 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{$ $}  %My kludge to get the first page to move down a bit
\vskip.75truein
\centerline{\titlefont Derivatives}
\bigskip
\centerline{\rm Arun Ram}
%${}^\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\noindent
A {\bf function} eats a number, chews on it, and spits out another
number.
$$PICTURE$$
A {\bf constant function} always spits out the same number,
no matter what the input is.

\bigskip\noindent
{\bf Example:} $f(x)=2$.

$$PICTURE$$
We call this function $2$.
\smallskip\noindent
So, 2 {\it sometimes means the number $2$}, and
{\it sometimes means the function $2$}.

\bigskip\noindent
A {\bf derivative} eats a function, chews on it, and spits out another
function.  
$$PICTURE$$
The derivative $\displaystyle{d\over dx}$ knows what to spit out by 
always following the rules:
\smallskip\noindent
\itemitem{(1)} $\displaystyle{ {dx\over dx} = 1},$
\smallskip\noindent
\itemitem{(2)} $\displaystyle{ {d(cf)\over dx} = c {df\over dx}},$
\qquad if $c$ does not change when $x$ changes,
\smallskip\noindent
\itemitem{(3)} $\displaystyle{ {d(f+g)\over dx} = 
{df\over dx} + {dg\over dx}},$
\smallskip\noindent
\itemitem{(4)} $\displaystyle{ {d(fg)\over dx} = 
f\;{dg\over dx} + {df\over dx}\;g}.$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=5x$.
\medskip
$\displaystyle{
{dy\over dx} = {d(5x)\over dx} = 5\; {dx\over dx} = 5\cdot 1 = 5. }$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=\pi x$.
\medskip
$\displaystyle{
{dy\over dx} = {d(\pi x)\over dx} = \pi\; {dx\over dx} = \pi\cdot 1 = \pi. }$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=1$.
\medskip
$$
{dy\over dx} = {d1\over dx} = {d(1\cdot 1)\over dx} 
= 1\cdot{d1\over dx}+ {d1\over dx}\cdot 1
= {d1\over dx}+{d1\over dx}. 
$$
Subtract $\displaystyle{d1\over dx}$ from both sides.
\smallskip\noindent
So \qquad $\displaystyle{ {d1\over dx} = 0 }.$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=5$.
\medskip
$$
{dy\over dx} = {d5\over dx} = {d(5\cdot 1)\over dx} 
= 5\cdot{d1\over dx} = 5\cdot 0 = 0.
$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=6342$.
\medskip
$$
{dy\over dx} = {d\,6342\over dx} = {d(6342\cdot 1)\over dx} 
= 6342\cdot{d1\over dx} = 6342\cdot 0 = 0.
$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dc\over dx}}$ if $c$ is a 
constant.
\medskip
$$
{dc\over dx} = {d(c\cdot 1)\over dx} 
= c\cdot{d1\over dx} = c\cdot 0 = 0.
$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=3x+12$.
\medskip
$$
{dy\over dx} 
= {d(3x+12)\over dx} 
= {d(3x)\over dx} + {d(12)\over dx} 
= 3{dx\over dx} + 0
= 3\cdot 1 + 0 = 3.
$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=x^2$.
\medskip
$$
{dy\over dx} 
= {dx^2\over dx} 
= {d(x\cdot x)\over dx} 
= x\,{dx\over dx} + {dx\over dx}\,x 
= x\cdot 1+1\cdot x 
=2x.
$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=x^3$.
\medskip
$$
{dy\over dx} 
= {dx^3\over dx} 
= {d(x^2\cdot x)\over dx} 
= x^2\,{dx\over dx} + {dx^2\over dx}\,x 
= x^2\cdot 1+2x\cdot x 
=3x^2.
$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=x^4$.
\medskip
$$
{dy\over dx} 
= {dx^4\over dx} 
= {d(x^3\cdot x)\over dx} 
= x^3\,{dx\over dx} + {dx^3\over dx}\,x 
= x^3\cdot 1+3x^2\cdot x 
=4x^3.
$$

\bigskip\noindent
$\ldots$ and we keep on going $\ldots$
\bigskip

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dy\over dx}}$ if $y=x^{6342}$.
\medskip
$$
{dy\over dx} 
= {dx^{6342}\over dx} 
= {d(x^{6341}\cdot x)\over dx} 
= x^{6341}\,{dx\over dx} + {dx^{6341}\over dx}\,x 
= x^{6341}\cdot 1+{6341}x^{6340}\cdot x 
=6342x^{6341}.
$$

\bigskip\noindent
$\ldots$ and we keep on going $\ldots$
\bigskip

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dx^n\over dx}}$ for $n=1,2,3,\ldots$.
\medskip
$$\eqalign{
{dy\over dx} 
&= {dx^n\over dx} 
= {d(x^{n-1}\cdot x)\over dx}  
= x^{n-1}\,{dx\over dx} + {dx^{n-1}\over dx}\,x  \cr
&= x^{n-1}\cdot 1+(n-1)x^{n-2}\cdot x,
\qquad\qquad\hbox{since we already found}\ \ {d x^{n-1}\over dx} = 
(n-1)x^{n-2}, \cr
&=nx^{n-1}. \cr
}
$$

\bigskip\noindent
and thus we have found \quad 
$\displaystyle{ {dx^n\over dx} = n x^{n-1},}$
\quad for all positive integers $n$. (Amazing!)

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dx^n\over dx}}$ for $n=0$.
\medskip
$$
{dy\over dx} 
= {dx^0\over dx} 
= {d1\over dx}  =0 = 0x^{-1} = 0 x^{0-1}.
$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dx^{-6342}\over dx}}$.
\medskip
$$
{d x^{-6342}\cdot x^{6342}\over dx} 
= {dx^0\over dx} 
= {d1\over dx}  =0.$$
On the other hand,
$$
{d x^{-6342}\cdot x^{6342}\over dx} 
= x^{-6342} {dx^{6342}\over dx} 
+ {dx^{-6342}\over dx}\cdot x^{6342}
= x^{-6342}\cdot 6342x^{6341}
+ {dx^{-6342}\over dx}\cdot x^{6342}.$$
So
$$0 
= x^{-6342}\cdot 6342x^{6341}
+ {dx^{-6342}\over dx}\cdot x^{6342}.$$
Solve for 
$\displaystyle{{dx^{-6342}\over dx}}$.
$${dx^{-6342}\over dx} = 
-6342 x^{-1}x^{-6342} = (-6342)x^{-6343}.$$

\bigskip\noindent
{\bf Example:} Find $\displaystyle{{dx^{-n}\over dx}}$ for $n=1,2,3,\ldots$.
\medskip
$$
{d x^{-n}\cdot x^n\over dx} 
= {dx^0\over dx} 
= {d1\over dx}  =0.$$
On the other hand,
$$
{d x^{-n}\cdot x^n\over dx} 
= x^{-n} {dx^n\over dx} 
+ {dx^{-n}\over dx}\cdot x^n
= x^{-n}\cdot nx^{n-1}
+ {dx^{-n}\over dx}\cdot x^n.$$
So
$$0 = x^{-n}\cdot nx^{n-1}
+ {dx^{-n}\over dx}\cdot x^n.$$
Solve for 
$\displaystyle{{dx^{-n}\over dx}}$.
$${dx^{-n}\over dx} = 
-n x^{-1}x^{-n} = (-n)x^{-n-1}.$$

\bigskip\noindent
and thus we have found \quad 
$\displaystyle{ {dx^n\over dx} = n x^{n-1},}$
\quad for {\it all} integers $n$. (AMAZING!)

\bigskip\noindent
{\bf Example:} Let $y=3x^3+5x^2+2x+7$.  Find $\displaystyle{{dy\over dx}}$.
\medskip
$$\eqalign{
{dy\over dx} 
&= {d(3x^3+5x^2+2x+7)\over dx} \cr
&= {d(3x^3)\over dx} + {d(5x^2+2x+7)\over dx}  \cr
&= {d(3x^3)\over dx} + {d(5x^2)\over dx} + {d(2x+7)\over dx}  \cr
&= {d(3x^3)\over dx} + {d(5x^2)\over dx} + {d(2x)\over dx} 
+{d7\over dx}  \cr
&= 3\,{dx^3\over dx} + 5\,{dx^2\over dx} + 2\,{dx\over dx} 
+7\,{d1\over dx}  \cr
&= 3\cdot 3x^2 + 5\cdot 2x + 2\cdot 1 +7\cdot 0
= 9x^2 + 10x + 2. \cr
}
$$

\bigskip\noindent
{\bf Example:} Let $y=-7x^{-13}+5x^{-7}+(6+2i)x^{38}$.  
Find $\displaystyle{{dy\over dx}}$.
\medskip
$$\eqalign{
{dy\over dx} 
&= {d(-7x^{-13}+5x^{-7}+(6+2i)x^{38})\over dx} \cr
&= {d(-7x^{-13})\over dx} + {d(5x^{-7})\over dx} + {d((6+2i)x^{38})\over dx} 
\cr
&= -7\,{dx^{-13}\over dx} + 5\,{dx^{-7}\over dx} + (6+2i)\,{dx^{38}\over dx} 
\cr
&= (-7)\cdot (-13)x^{-13-1}+5(-7)x^{-7-1}+(6+2i)\cdot 38\cdot x^{38-1} \cr
&= 91x^{-14}-35x^{-8}+(228+76i)x^{37}. \cr
}
$$


\vfill\eject
\end