(%i1) Cornelsen, S. 89, Nr. 2

(%i2) "a)"$

(%i3) f(x) := (2*x - 3) / (x + 5);
                                        2 x - 3
(%o3)                           f(x) := -------
                                         x + 5

(%i4) diff((2*x - 3) / (x + 5),x);
                                 2     2 x - 3
(%o4)                          ----- - --------
                               x + 5          2
                                       (x + 5)

(%i5) sqfr(%);
                                      13
(%o5)                              --------
                                          2
                                   (x + 5)

(%i6) diff((2*x - 3) / (x + 5),x,2);
                            2 (2 x - 3)      4
(%o6)                       ----------- - --------
                                    3            2
                             (x + 5)      (x + 5)

(%i7) sqfr(%);
                                       26
(%o7)                             - --------
                                           3
                                    (x + 5)

(%i8) "b)"$

(%i9) f(x) := (x) / (1-x);
                                           x
(%o9)                            f(x) := -----
                                         1 - x

(%i10) diff((x) / (1-x),x,1);
                                  x         1
(%o10)                         -------- + -----
                                      2   1 - x
                               (1 - x)

(%i11) sqfr(%);
                                      1
(%o11)                             --------
                                          2
                                   (x - 1)

(%i12) diff((x) / (1-x),x,2);
                                2 x         2
(%o12)                        -------- + --------
                                     3          2
                              (1 - x)    (1 - x)

(%i13) sqfr(%);
                                       2
(%o13)                            - --------
                                           3
                                    (x - 1)

(%i14) "c)"$

(%i15) f(x) := (x^3 + 2*x) / (x+3);
                                        3
                                       x  + 2 x
(%o15)                         f(x) := --------
                                        x + 3

(%i16) diff((x^3 + 2*x) / (x+3),x,1);
                                 2        3
                              3 x  + 2   x  + 2 x
(%o16)                        -------- - --------
                               x + 3            2
                                         (x + 3)

(%i17) sqfr(%);
                                   3      2
                                2 x  + 9 x  + 6
(%o17)                          ---------------
                                          2
                                   (x + 3)

(%i18) diff((x^3 + 2*x) / (x+3),x,2);
                                  3                2
                       6 x    2 (x  + 2 x)   2 (3 x  + 2)
(%o18)                ----- + ------------ - ------------
                      x + 3            3              2
                                (x + 3)        (x + 3)

(%i19) sqfr(%);
                               3      2
                           2 (x  + 9 x  + 27 x - 6)
(%o19)                     ------------------------
                                          3
                                   (x + 3)

(%i20) "d)"$

(%i21) f(x) := (x^3) / (x-1);
                                           3
                                          x
(%o21)                           f(x) := -----
                                         x - 1

(%i22) diff((x^3)/(x-1),x,1);
                                  2        3
                               3 x        x
(%o22)                         ----- - --------
                               x - 1          2
                                       (x - 1)

(%i23) sqfr(%);
                                  2
                                 x  (2 x - 3)
(%o23)                           ------------
                                          2
                                   (x - 1)

(%i24) diff((x^3)/(x-1),x,2);
                               3          2
                            2 x        6 x       6 x
(%o24)                    -------- - -------- + -----
                                 3          2   x - 1
                          (x - 1)    (x - 1)

(%i25) sqfr(%);
                                    2
                              2 x (x  - 3 x + 3)
(%o25)                        ------------------
                                          3
                                   (x - 1)

(%i26) "e)"$

(%i27) f(x) := (1 + x^0.5) / (1 - x^0.5);
                                            0.5
                                       1 + x
(%o27)                         f(x) := --------
                                            0.5
                                       1 - x

(%i28) diff((1 + x^0.5)/(1 - x^0.5),x,1);
                             0.5
                       0.5 (x    + 1)          0.5
(%o28)                ---------------- + ---------------
                            0.5 2  0.5         0.5   0.5
                      (1 - x   )  x      (1 - x   ) x

(%i29) sqfr(%);

                                      1
(%o29)                         ----------------
                                 0.5     2  0.5
                               (x    - 1)  x

(%i30) diff((1 + x^0.5)/(1 - x^0.5),x,2);
              0.5                                   0.5
        0.5 (x    + 1)          0.5          0.25 (x    + 1)         0.25
(%o30) ---------------- + ---------------- - ---------------- - ---------------
             0.5 3  1.0         0.5 2  1.0         0.5 2  1.5         0.5   1.5
       (1 - x   )  x      (1 - x   )  x      (1 - x   )  x      (1 - x   ) x

(%i31) sqfr(%);


                                    3      1
                                   ---- - ----
                                    1.0    1.5
                                   x      x
(%o31)                          - -------------
                                      0.5     3
                                  2 (x    - 1)

(%i32) "f)"$

(%i33) f(x) := (3 * x^2 - x + 2) / (x^2 + x - 1);
                                        2
                                     3 x  - x + 2
(%o33)                       f(x) := ------------
                                       2
                                      x  + x - 1

(%i34) diff((3 * x^2 - x + 2)/(x^2 + x - 1),x,1);
                                                2
                      6 x - 1     (2 x + 1) (3 x  - x + 2)
(%o34)               ---------- - ------------------------
                      2                  2         2
                     x  + x - 1        (x  + x - 1)

(%i35) sqfr(%);
                                   2
                                4 x  - 10 x - 1
(%o35)                          ---------------
                                   2         2
                                 (x  + x - 1)

(%i36) diff((3 * x^2 - x + 2)/(x^2 + x - 1),x,2);
                                                  2
           6        2 (2 x + 1) (6 x - 1)   2 (3 x  - x + 2)
(%o36) ---------- - --------------------- - ----------------
        2                 2         2          2         2
       x  + x - 1       (x  + x - 1)         (x  + x - 1)
                                                               2     2
                                                    2 (2 x + 1)  (3 x  - x + 2)
                                                  + ---------------------------
                                                             2         3
                                                           (x  + x - 1)

(%i37) sqfr(%);
                                 3       2
                           2 (4 x  - 15 x  - 3 x - 6)
(%o37)                   - --------------------------
                                   2         3
                                 (x  + x - 1)