Derivatives of Functions

Maple can differentiate most elementary functions:
1. The differentiation command is diff:
> diff(3*x^4 -4*x^2 -5, x);

                                 3
                             12 x  - 8 x

> diff((x+1)^2/(x^2 +2*x)^2, x);

                                          2
                      x + 1        (x + 1)  (2 x + 2)
                 2 ----------- - 2 ------------------
                     2       2          2       3
                   (x  + 2 x)         (x  + 2 x)

Investigating a Function Using plot and diff
> f := (x+2)/(3 +(x^2 +1)^3);

                                   x + 2
                          f := -------------
                                     2     3
                               3 + (x  + 1)

> plot(f, x=-5..5);

> plot(f, x=-5..5, y=-0.1..0.1);

You can determine where maxima,minima, and points of inflection occur by
looking at the derivatives of f(x):
> f:= (x+2)/(3+(x^2 +1)^3);

                                   x + 2
                          f := -------------
                                     2     3
                               3 + (x  + 1)

> d := diff( f,x);

                                               2     2
                         1           (x + 2) (x  + 1)  x
              d := ------------- - 6 -------------------
                         2     3             2     3 2
                   3 + (x  + 1)       (3 + (x  + 1) )

> simplify(d);

                     6      4      2       5       3
             -4 + 5 x  + 9 x  + 3 x  + 12 x  + 24 x  + 12 x
           - ----------------------------------------------
                              6      4      2 2
                        (4 + x  + 3 x  + 3 x )

> fsolve(numer("));

                      -2.485165927, .2700964050

You can more clearly see the behavior of the function to the left of zero
by choosing appropriate x and y values:
> plot(f, x=-4..0, y=-0.01..0.01);

Now for the points of inflection:
> diff(d, x);

        2     2                     2     4  2
      (x  + 1)  x         (x + 2) (x  + 1)  x
-12 ---------------- + 72 --------------------
           2     3 2               2     3 3
    (3 + (x  + 1) )         (3 + (x  + 1) )

                    2       2               2     2
          (x + 2) (x  + 1) x      (x + 2) (x  + 1)
     - 24 ------------------- - 6 -----------------
                  2     3 2              2     3 2
           (3 + (x  + 1) )        (3 + (x  + 1) )

> simplify(");

    2                       2       4       6      5       3      9
6 (x  + 1) (-8 - 12 x - 22 x  + 36 x  + 40 x  + 6 x  - 25 x  + 5 x

           7       8    /       6      4      2 3
     + 12 x  + 14 x )  /  (4 + x  + 3 x  + 3 x )
                      /

> fsolve(numer("));

               -2.964270928, -.5472819999, .8702050673

Integrals of Functions

> int(3*x^4 -2*x,x);

                                  5    2
                             3/5 x  - x

> int(sec(x)^4,x);

                           sin(x)        sin(x)
                       1/3 ------- + 2/3 ------
                                 3       cos(x)
                           cos(x)

Maple can perform integration by parts:
> int(x^3*ln(x), x);

                             4               4
                        1/4 x  ln(x) - 1/16 x

> int(x^2/sqrt(x^2 -9), x);

                     2     1/2                2     1/2
             1/2 x (x  - 9)    + 9/2 ln(x + (x  - 9)   )

> diff(", x);

                                                        x
                                               1 + -----------
                                  2                  2     1/2
             2     1/2           x                 (x  - 9)
       1/2 (x  - 9)    + 1/2 ----------- + 9/2 ---------------
                               2     1/2             2     1/2
                             (x  - 9)          x + (x  - 9)

> simplify(");

                                  2
                                 x
                             -----------
                               2     1/2
                             (x  - 9)

> int((x^2 +2*x +1)/((x^2 +1)^2*(x-2)), x);

                             2        13                  -2 x - 4
   9/25 ln(x - 2) - 9/50 ln(x  + 1) - -- arctan(x) - 1/10 --------
                                      25                    2
                                                           x  + 1

You can also evaluate definite integrals:
> int(sin(x), x=0..Pi/2);

                                  1

> with(student);

[D, Diff, Doubleint, Int, Limit, Lineint, Product, Sum, Tripleint,

    changevar, combine, completesquare, distance, equate, extrema,

    integrand, intercept, intparts, isolate, leftbox, leftsum,

    makeproc, maximize, middlebox, middlesum, midpoint, minimize,

    powsubs, rightbox, rightsum, showtangent, simpson, slope,

    trapezoid, value]

> Int(x*sin(x), x);

                             /
                            |
                            |  x sin(x) dx
                            |
                           /

The Int operator with a capital I creates the integral but does not evaluate it.
> intparts(",x);

                                /
                               |
                     intparts( |  x sin(x) dx, x)
                               |
                              /

>