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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.
In[1]:=
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Out[1]=
Here is a slightly more complicated example.
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Click for copyable input
Out[2]=
D[f,x]the (partial) derivative
Integrate[f,x]the indefinite integral Integralf dx
Sum[f,{i,imin,imax}]the sum f
Solve[lhsEqualrhs,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.