The parametric equations for    for    found in most text books are

    ,  
and
    .  
    
We now derive these equations using Mathematica.

Derivation.

Remark. If you have stored anything in  , , , or with subscripts, you will need to Remove them, which is stronger than  Clear them. This is complicated, so in doubt do the following.  
Warning.  Do not  Remove a variable that does not exist.  

Proceed with the derivation.  

There are eight coefficients which must be computed.  So we will need eight equations.  
The following two equations force the Bézier curve to pass through the starting point

The next two equations force the Bézier curve to pass through the destination point

The following two equations force the Bézier curve to have the desired tangent vector at pointing in the direction of the control point .

The next two equations force the Bézier curve to have the desired tangent vector at pointing in the direction of the control point .

We now use Mathematica to solve the eight equations for the eight coefficients required to form the two cubic equations.
Aside. The reason we introduce the factor will be apparent when we look at the solution using Bernstein polynomials.  

This is equivalent to the formula we wanted to establish.  

To verify we have the correct end point conditions, we can evaluate    and its derivative at and ,  and see if it has the required properties.  

The function    and its derivative at and , also have the required properties.  

(c) John H. Mathews 2003