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) Apply the product rule to get:
$\dfrac{d}{dt}(u \cdot ( v \times w)=u'\cdot (v \times w)+u \cdot (v \times w'+v' \times w)$
or, $=u'\cdot (v \times w)+u \cdot (v \times w')+u \cdot (v' \times w)$
b) Apply product rule to get:
$\dfrac{d}{dt}(r \cdot ( r' \times r'')=r \cdot (r'' \times r''+ r' \times r''')+0$
or, $=r \cdot (r' \times r''')$
Hence, 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) $\dfrac{d}{dt}(r \cdot ( r' \times r'')=r \cdot (r' \times r''')$
