Answer
$$\frac{d}{dt}(r(t)\cdot r_1(t))|_2
=62$$
Work Step by Step
Since $$ r_1(t)=\langle t^2,t^3,t \rangle $$
then, we have
$$\frac{d}{dt}(r(t)\cdot r_1(t))=\frac{d}{dt}(r(t))\cdot r_1(t) +r(t)\cdot \frac{d}{dt}( r_1(t))\\
=\frac{d}{dt}(r(t))\cdot r_1(t) +r(t)\cdot \langle 2t,3t^2,1\rangle.$$
Now, at $ t=2$, we have
$$\frac{d}{dt}(r(t)\cdot r_1(t))|_2=\frac{d}{dt}(r(2))\cdot r_1(2) +r(2)\cdot \frac{d}{dt}( r_1(2))\\
=\langle 1,4,3 \rangle\cdot \langle 4,8,2 \rangle+\langle 2,1,0 \rangle\cdot \langle 4,12,1 \rangle\\
=62$$
