This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.2)

Symbolic Calculations

Mathematica's ability to deal with symbolic expressions, as well as numbers, allows you to use it for many kinds of mathematics.
Calculus is one example. With Mathematica, you can differentiate an expression symbolically, and get a formula for the result.
This finds the derivative of xn.
Click for copyable input
Here is a slightly more complicated example.
Click for copyable input
D[f,x]the (partial) derivative
Integrate[f,x]the indefinite integral f dx
Sum[f,{i,imin,imax}]the sum f
Solve[lhsrhs,x]solution to an equation for x
Series[f,{x,x0,order}]a power series expansion of f about the point x=x0
Limit[f,x->x0]the limit
Minimize[f,x]minimization of f with respect to x

Some symbolic mathematical operations.

Getting formulas as the results of computations is usually desirable when it is possible. There are however many circumstances where it is mathematically impossible to get an explicit formula as the result of a computation. This happens, for example, when you try to solve an equation for which there is no "closed form" solution. In such cases, you must resort to numerical methods and approximations.