Some of the best known symbolic mathematics applications are Mathematica, Maple and Derive. This last one, is one of the most used in university environments due to how easy it is to use. It's a powerful mathematics program developed to operate with symbolic notation, even though it also allows numerical computation.
Derive is capable of solving derivatives, integrals, limits, series, vector and matrix operations, polynomials, algebraic fractions,... Furthermore, it has several options for the graphical representation of bidimensional and tridimensional functions. The result of the operations performed with Derive will be an exact value, expressed with a finite number of decimals, with the precision chosen by the user.
If you're looking for an intuitive application to learn mathematics, download Derive.