# 4.1  Implicit Differentiation
# Art Belmonte, Summer 1996
# These are the shaded gray examples which appear in Section 4.1.
# In-line plots have been removed to conserve disk space.
> restart; with(plots): eq:=3*x*y = x^3 + y^3;

                                       3    3
                        eq := 3 x y = x  + y

> implicitplot(eq, x=-3..3, y=-3..3, scaling=constrained);
> x:=1.5; fsolve(eq, y=1..2);

                               x := 1.5


                             1.500000000

> x:='x'; implicitplot(eq, x=1.25..1.75,
>      y=1.25..1.75, scaling=constrained);

                                x := x

> implicitplot(eq, x=1.49..1.51,
>      y=1.49..1.51, scaling=constrained);
> eq:=3*x*y = x^3 + y^3; subs(y=y(x), ");

                                       3    3
                        eq := 3 x y = x  + y


                                    3       3
                        3 x y(x) = x  + y(x)

> diff(", x); solve(", diff(y(x), x));

                       /d      \      2         2 /d      \
          3 y(x) + 3 x |-- y(x)| = 3 x  + 3 y(x)  |-- y(x)|
                       \dx     /                  \dx     /


                                      2
                              y(x) - x
                              ----------
                                       2
                              -x + y(x)

> subs(y(x)=y, "); Dy:=simplify(");

                                    2
                               y - x
                               -------
                                     2
                               -x + y


                                       2
                                  y - x
                            Dy := -------
                                        2
                                  -x + y

> m:=subs(x=3/2, y=3/2, Dy);

                               m := -1

>