Answer
a) $\dfrac{d}{dt}(u \cdot ( v \times w)=u'\cdot (v \times w)+u \cdot (v \times w')+u \cdot (v' \times w)$
b) $\dfrac{d}{dt}(r \cdot ( r' \times r'')=r \cdot (r' \times r''')$
Work Step by Step
a. Need to use product rule such as: $\dfrac{d}{dt}(u \cdot ( v \times w)=u'\cdot (v \times w)+u \cdot (v \times w'+v' \times w)$
or, $=u'\cdot (v \times w)+u \cdot (v \times w')+u \cdot (v' \times w)$
Therefore, $\dfrac{d}{dt}(u \cdot ( v \times w)=u'\cdot (v \times w)+u \cdot (v \times w')+u \cdot (v' \times w)$
b. Need to use product rule such as: $\dfrac{d}{dt}(r \cdot ( r' \times r'')=r \cdot (r'' \times r''+ r' \times r''')+(0)$
Therefore, $\dfrac{d}{dt}(r \cdot ( r' \times r'')=r \cdot (r' \times r''')$
