Examples of time derivatives of vector expressions

vector_time_derivatives.txt

Let the vector v be a function of time and k be a constant vector. Find the time derivatives of 

1. |v|^2

   d |v|^2 = d  (v⋅v)
   dt        dt
           = dv⋅v + v⋅dv
             dt       dt
           = 2v⋅dv
                dt

2. (v⋅k)v

   d (v⋅k)v = (d (v⋅k))v + (v⋅k)dv
   dt          dt               dt
            = (dv⋅k + v⋅dk)v + (v⋅k)dv
               dt       dt          dt
            = (dv⋅k)v + (v⋅k)dv
               dt            dt

3. [v, dv, k] (This is the triple scalar product of three vectors)
       dt

   d [v, dv, k] = d (v⋅dv⨯k)
   dt    dt       dt   dt
                = dv⋅(dv⨯k) + v⋅d (dv⨯k)
                  dt  dt        dt dt
                = dv⋅dv⨯k + v⋅(d (dv)⨯k + dv⨯dk)
                  dt dt        dt dt      dt dt
                = v⋅d (dv)⨯k
                    dt dt
                = [v, d (dv), k]
                      dt dt