Calc 3 practice test question possibly involving chain rule and cross product? I have my final in a few hours, please please help.

Show that if F"(t) exists for all t then:
d/dt [F(t) X F'(t)] = F(t) X F"(t)
It is 

$$\frac{d}{dt}(u(t)\times v(t))=u'(t)\times v(t)+u(t)\times v'(t).$$ Thus

$$\frac{d}{dt}(F(t)\times F'(t))=F'(t)\times F'(t)+F(t)\times F''(t).$$ Now, we have that $$F'(t)\times F'(t)=0.$$ Thus

$$\frac{d}{dt}(F(t)\times F'(t))=F(t)\times F''(t).$$
It is 

$$\frac{d}{dt}(u(t)\times v(t))=u'(t)\times v(t)+u(t)\times v'(t).$$ Thus

$$\frac{d}{dt}(F(t)\times F'(t))=F'(t)\times F'(t)+F(t)\times F''(t).$$ Now, we have that $$F'(t)\times F'(t)=0.$$ Thus

$$\frac{d}{dt}(F(t)\times F'(t))=F(t)\times F''(t).$$