% This file:
%   http://anggtwu.net/LATEX/2025-1-C2-S-defs.tex.html
%   http://anggtwu.net/LATEX/2025-1-C2-S-defs.tex
%          (find-angg "LATEX/2025-1-C2-S-defs.tex")
% Author: Eduardo Ochs <eduardoochs@gmail.com>
%
% «.P»			(to "P")
% «.DefDeriv»		(to "DefDeriv")
% «.TFC2»		(to "TFC2")
% «.DFI»		(to "DFI")
% «.MV:reset»		(to "MV:reset")
% «.MV»			(to "MV")
% «.EDOVS:reset»	(to "EDOVS:reset")
% «.EDOVS»		(to "EDOVS")
% «.Aipim»		(to "Aipim")
% «.eq-if»		(to "eq-if")
% «.p-if»		(to "p-if")



% «P»  (to ".P")
\def\P   #1{\left( {#1} \right)}
\def\Pbig#1{ \big( {#1}   \big)}
\def\PBig#1{ \Big( {#1}   \Big)}

% «DefDeriv»  (to ".DefDeriv")
% (c2m251stp 3 "DefDeriv")
% (c2m251sta   "DefDeriv")
% 2kT63: (c2m251tnp 4 "defs-DD")
%        (c2m251tna   "defs-DD")
% (find-angg "MAXIMA/2025-1-s.mac" "TFC2")
\sa  {DDL}        {\left. \ddx f(x) \, \right|_{x=a}}
\sa {[DD1]}       {\CFname{DD1}{}}
\sa {[DD2]}       {\CFname{DD2}{}}
\sa {(DD1)}       {\P{\D \ga{DDL} = \lim_{ε→0} \frac{f(x+ε)-f(x)}{ε}}}
\sa {(DD2)}       {\P{\D \ga{DDL} = \lim_{a→x} \frac{f(a)-f(x)}{x-a}}}
\sa {[DefAt]}     {\CFname{DefAt}{}}
\sa {(DefAt)}     {\PBig{\left. f(x) \right|_{x=a} \;=\; f(a)}}
\sa {[DefDeriv1]} {\CFname{DefDeriv1}{}}
\sa {[DefDeriv2]} {\CFname{DefDeriv2}{}}
\sa {(DefDeriv1)} {\P{\D \ga{DDL} = \lim_{ε→0} \frac{f(x+ε)-f(x)}{ε}}}
\sa {(DefDeriv2)} {\P{\D \ga{DDL} = \lim_{a→x} \frac{f(a)-f(x)}{x-a}}}

% «TFC2»  (to ".TFC2")
% (c2m251stp 3 "TFC2")
% (c2m251sta   "TFC2")
% (c2m251sda   "TFC2")
\sa  {[DefDif]}{\CFname{DefDif}{}}
\sa  {(DefDif)}{\PBig{\D \difx{a}{b}{F(x)} \;=\; F(b)-F(a)}}
\sa  {[TFC2]}  {\CFname{TFC2}{}}
\sa  {(TFC2)}  {\P{\D \Intx{a}{b}{F'(x)} \;=\; \difx{a}{b}{F(x)}}}
\sa  {[II]}    {\CFname{II}{}}
\sa  {(II)}    {\P{\D \intx{F'(x)} \;=\; F(x)}}
\sa  {[RDC]}   {\CFname{RDC}{}}
\sa  {(RDC)}   {\PBig{\ddx c \;=\; 0}}
\sa  {[RMC]}   {\CFname{RMC}{}}
\sa  {(RMC)}   {\PBig{\ddx(cf(x)) \;=\; c\ddx f(x)}}
\sa  {[RPot]}  {\CFname{RPot}{}}
\sa  {(RPot)}  {\PBig{\ddx x^n \;=\; nx^{n-1}}}
\sa  {[RSoma]} {\CFname{RSoma}{}}
\sa  {(RSoma)} {\PBig{\ddx(f(x)+g(x)) \;=\; \ddx f(x) + \ddx g(x)}}
\sa  {[RProd]} {\CFname{RProd}{}}
\sa  {(RProd)} {\PBig{\ddx(f(x)g(x)) \;=\; f(x) \ddx g(x) + g(x) \ddx f(x)}}
\sa  {[RC]}    {\CFname{RC}{}}
\sa  {(RC)}    {\PBig{\ddx f(g(x)) \;=\; f'(g(x))g'(x)}}

% «DFI»  (to ".DFI")
% (c2m251stp 4 "DFI")
% (c2m251sta   "DFI")
\sa {[DFI]} {\CFname{DFI}{}}
\sa {(DFI)} {\P{
  \begin{array}{rcl}
    f(g(x))       &=& x \\
    \ddx f(g(x))  &=& \ddx x \\
                  &=& 1 \\
    \ddx f(g(x))  &=& f'(g(x))g'(x) \\
    f'(g(x))g'(x) &=& 1 \\
    g'(x)         &=& \D \frac{1}{f'(g(x))} \\
  \end{array}}}


% «MV:reset»  (to ".MV:reset")
% (c2m251sta     "MV-defaults")
% (c2m251mvdefsa "mv-defaults")
% (c2m241dip 7 "MVDs-e-MVIs-color")
% (c2m241dia   "MVDs-e-MVIs-color")
% \def\MVf #1{f (#1)}
% \def\MVfp#1{f'(#1)}
% \def\MVg #1{g (#1)}
% \def\MVgp#1{g'(#1)}
\sa{MV:reset}{
  \sa          {.fg}{}
  \sa            {a}{a}
  \sa            {b}{b}
  \sa            {u}{u}
  \sa            {x}{x}
  \sa         {f(u)}{f (\ga{u})}
  \sa         {g(a)}{g (\ga{a})}
  \sa         {g(b)}{g (\ga{b})}
  \sa         {g(x)}{g (\ga{x})}
  \sa        {f'(u)}{f'(\ga{u})}
  \sa        {g'(x)}{g'(\ga{x})}
  \sa      {f(g(a))}{f (\ga{g(a)})}
  \sa      {f(g(b))}{f (\ga{g(b)})}
  \sa      {f(g(x))}{f (\ga{g(x)})}
  \sa     {f'(g(x))}{f'(\ga{g(x)})}
  \sa{f'(g(x))g'(x)}{\ga{f'(g(x))} \ga{.fg} \ga{g'(x)}}
  }
\ga{MV:reset}

% «MV»  (to ".MV")
\sa {[MVD4]} {\CFname{MVD4}{}}
\sa {(MVD4)} {\P{\begin{array}{rcl}\ga{MVD4}\end{array}}}
\sa  {MVD4}  {
        \D \Intx {\ga{a}} {\ga{b}} {\ga{f'(g(x))g'(x)}}
    &=& \D \Difx {\ga{a}} {\ga{b}} {\ga{f(g(x))}}       \\
    &=& \D                \ga{f(g(b))} - \ga{f(g(a))}   \\
    &=& \D \Difu {\ga{g(a)}} {\ga{g(b)}}  {\ga{f(u)}}   \\
    &=& \D \Intu {\ga{g(a)}} {\ga{g(b)}} {\ga{f'(u)}}   \\
  }

\sa {[MVD1]} {\CFname{MVD1}{}}
\sa {(MVD1)} {\P{\begin{array}{rcl}\ga{MVD1}\end{array}}}
\sa  {MVD1}  {
        \D \Intx {\ga{a}} {\ga{b}} {\ga{f'(g(x))g'(x)}}
    &=& \D \Intu {\ga{g(a)}} {\ga{g(b)}} {\ga{f'(u)}}   \\
  }

\sa {[MVI3]} {\CFname{MVI3}{}}
\sa {(MVI3)} {\P{\begin{array}{rcl}\ga{MVI3}\end{array}}}
\sa  {MVI3}  {
        \D \intx {\ga{f'(g(x))g'(x)}}
    &=& \D        \ga{f(g(x))}        \\
    &=& \D        \ga{f(u)}           \\
    &=& \D \intu {\ga{f'(u)}}         \\
  }

\sa {[MVI1]} {\CFname{MVI1}{}}
\sa {(MVI1)} {\P{\begin{array}{rcl}\ga{MVI1}\end{array}}}
\sa  {MVI1}  {
        \D \intx {\ga{f'(g(x))g'(x)}}
    &=& \D \intu {\ga{f'(u)}}         \\
  }



% «EDOVS:reset»  (to ".EDOVS:reset")
\sa{EDOVS:reset}{
  \sa           {G(x)} {G(x)}
  \sa           {H(y)} {H(y)}
  \sa           {g(x)} {g(x)}
  \sa           {h(y)} {h(y)}
  \sa        {Hinv(u)} {H^{-1}(u)}
  \sa     {Hinv(H(y))} {H^{-1}(H(y))}
  \sa {Hinv(G(x)+C_3)} {H^{-1}(G(x)+C_3)}
  }
\sa{EDOVS:reset-S1}{
  \sa           {g(x)} {-2x}
  \sa           {h(y)} {2y}
  \sa           {G(x)} {-x^2}
  \sa           {H(y)} {y^2}
  \sa        {Hinv(u)} {\sqrt{u}}
  \sa     {Hinv(H(y))} {\sqrt{y^2}}
  \sa {Hinv(G(x)+C_3)} {\sqrt{-x^2+C_3}}
  }
\ga{EDOVS:reset}


% «EDOVS»  (to ".EDOVS")
\sa {[M]} {\CFname{M}{}}
\sa {(M)} {\P{\begin{array}{rcl}\ga{M}\end{array}}}
\sa  {M}  {
             \D \dydx &=& \D \frac{\ga{g(x)}}{\ga{h(y)}} \\
        \ga{h(y)}\,dy &=& \ga{g(x)}\,dx       \\ \\[-10pt]
     \inty{\ga{h(y)}} &=& \intx{\ga{g(x)}}    \\
           \mcc{\veq} & & \mcc{\veq}          \\
  \mcc{\ga{H(y)}+C_1} & & \mcc{\ga{G(x)}+C_2} \\ \\[-10pt]
        \ga{H(y)}     &=& \ga{G(x)}+C_2-C_1   \\
                      &=& \ga{G(x)}+C_3       \\ \\[-10pt]
      \ga{Hinv(H(y))} &=& \ga{Hinv(G(x)+C_3)} \\
           \mcc{\veq} & & \\
              \mcc{y} & & \\
  }



\sa   {[F3]}{\CFname{F}{_3}}
\sa   {[F2]}{\CFname{F}{_2}}
\sa   {[S1]}{\CFname{S}{_1}}
\sa{(F3)}{
  \left(\begin{array}{rcl}
             \D \dydx &=& \D \frac{\ga{g(x)}}{\ga{h(y)}} \\ \\[-10pt]
      \ga{Hinv(H(y))} &=& \ga{Hinv(G(x)+C_3)} \\
           \mcc{\veq} & & \\
              \mcc{y} & & \\
   \end{array}
   \right)
  }
\sa{(F2)}{
  \left(\begin{array}{rcl}
             \D \dydx &=& \D \frac{\ga{g(x)}}{\ga{h(y)}} \\ \\[-10pt]
                    y &=& \ga{Hinv(G(x)+C_3)} \\
   \end{array}
   \right)
  }
\sa{(S)}{
  \left[\begin{array}{rcl}
             g(x) &:=& \ga{g(x)} \\
             h(y) &:=& \ga{h(y)} \\
             G(x) &:=& \ga{G(x)} \\
             H(y) &:=& \ga{H(y)} \\
        H^{-1}(u) &:=& \ga{Hinv(u)} \\
   \end{array}
   \right]
  }



% «Aipim»  (to ".Aipim")
% (c2m251stp 7 "Aipim")
% (c2m251sta   "Aipim")

% (c2m251introp 5 "defs-Aipim")
% (c2m251introa   "defs-Aipim")
\def\und#1#2{\underbrace{#1}_{#2}}
\def\setdepthto#1#2{\setbox1\hbox{$#2$}%
                    \dp1=#1%
                    \box1}
%
\sa{[Aipim]}   {\CFname{Aipim}{}}
\sa{(Aipim)}   {\Pbig{\sqrt{a^2+b^2} \;=\; a+b}}
\sa{(Aipim34)} {\Pbig{\sqrt{3^2+4^2} \;=\; 3+4}}
\sa {Aipim}    {\sqrt{a^2+b^2}=a+b}
\sa {Aipim u}{                                  % Aipim with "\und"s
  \sa{a 0}               {\setdepthto{2pt}{a}}
  \sa{b 0}               {\setdepthto{2pt}{b}}
  \sa{a}                 {\und{\ga{a 0}}{3}}
  \sa{b}                 {\und{\ga{b 0}}{4}}
  \sa{a^2}               {\und{\ga{a}^2}{9}}
  \sa{a^2}               {\und{{\ga{a}}^2}{9}}
  \sa{b^2}               {\und{{\ga{b}}^2}{16}}
  \sa{a^2+b^2 0}         {\und{\ga{a^2}+\ga{b^2}}{25}}
  \sa{a^2+b^2}           {\setdepthto{0pt}{\ga{a^2+b^2 0}}}
  \sa{sqrt(a^2+b^2) 0}   {\sqrt{\ga{a^2+b^2}}}
  \sa{sqrt(a^2+b^2) 1}   {\setdepthto{50pt}{\ga{sqrt(a^2+b^2) 0}}}
  \sa{sqrt(a^2+b^2)}     {\und{\ga{sqrt(a^2+b^2) 1}}{5}}
  \sa{a+b}               {\und{\ga{a}+\ga{b}}{7}}
  \sa{sqrt(a^2+b^2)=a+b} {\und{\ga{sqrt(a^2+b^2)} = \ga{a+b}}{\False}}
  \ga{sqrt(a^2+b^2)=a+b}
  }



% «eq-if»  (to ".eq-if")
% (c2m251stp  7 "eq-if")
% (c2m251sta    "eq-if")
% (c2m251sda    "eq-if")
\Sa         {=}#1{\sa{1}{#1}\ga{= after 1}}
\sa{=    after 1}{\ga{eq   after 1}}
\sa{=    after 1}{\ga{eqnp after 1}}
\sa{=    after 1}{\ifgaundefined{=.\ga{1}.}\ga{eq   after 1}
                  \else                    \ga{eqnp after 1}
                  \fi}
\Sa{eq   after 1}{=}
\Sa{eqnp after 1}{\overset{\scriptscriptstyle(\ga{1})}{=}}
\Sa{eqnp after 1}{\standout{$\overset{\scriptscriptstyle(\ga{1})}{=}$}}

% «p-if»  (to ".p-if")
% (c2m251stp 6 "p-if")
% (c2m251sta   "p-if")
% (c2m251sda   "p-if")
\Sa               {p}#1#2{\sa{1}{#1}\sa{2}{#2}\ga{p       after 2}}
\Sa{p       after 2}{\ifgaundefined{p.\ga{1}.}\ga{pnormal after 2}
                     \else                    \ga{pbox    after 2}
                     \fi}
\Sa{pnormal after 2}{\ga{2}}
\Sa{pbox    after 2}{\standout{$\ga{2}$}}




% Local Variables:
% coding: utf-8-unix
% ee-tla: "c2sd"
% ee-tla: "c2m251sd"
% End:
