Answer
$\frac{d}{dt}$(r(r' x r'')) = r(r' x r''')
Work Step by Step
$\frac{d}{dt}$(r(r' x r'')
Applying dot product rule
= r'(r' x r'') + r(r' x r'')'
r' is orthogonal to (r' x r'') by property of the cross product, thus:
= 0 + r(r'' x r'' + r' x r''')
Cross product of a vector to itself is 0, so r'' x r''=0
= r(r' x r''')
