Calculus
Mathematica TE can do calculus as well as algebra. Here is the derivative of 
.
In[1]:= D[Tan[x ^ 2], x]
Out[1]= 
This is the second derivative.
In[2]:= D[Tan[x ^ 2], {x, 2}]
Out[2]= 
This shows the tangent at 
to the curve 
.
In[3]:= Plot[{x^3, 12 (x - 2) + 8}, {x,-4, 4}]
Out[3]= 
Mathematica TE can evaluate the limits of many functions.
In[4]:= Limit[(1 + 2 / n) ^ n, n -> Infinity]
Out[4]= 
Here is an easy integral.
In[5]:= Integrate[ Tan[x], x ]
Out[5]= 
The integral of 
is a little harder.
In[6]:= Integrate[ x^2 Sin[x]^2, x ]
Out[6]= 
Differentiating the result gives the original integrand, but in a somewhat different form.
In[7]:= D[ %, x ]
Out[7]= 
The Mathematica TE function Simplify uses algebraic and trigonometric identities to simplify the expression, in this case recovering the original form.
In[8]:= Simplify[ % ]
Out[8]= 
This finds the power-series expansion of the previous result about the point 
.
In[9]:= Series[ %, {x, 0, 14} ]
Out[9]= 
