Answer
$$\frac{d}{dt}(r_1(t)\cdot r_2(t))
=(2t+3t^2)e^{3t}+(3t^2+2t^3)e^{2t}+(1+t)e^{t}.$$
Work Step by Step
Since $$ r_1(t)=\langle t^2,t^3,t \rangle, \quad r_2(t)=\langle e^{3t},e^{2t},e^{ t} \rangle $$
then, we have
$$\frac{d}{dt}(r_1(t)\cdot r_2(t))=\frac{d}{dt}(r_1(t))\cdot r_2(t) +r_1(t)\cdot \frac{d}{dt}( r_2(t))\\
=\langle 2 t,3t^2,1 \rangle\cdot \langle e^{3t},e^{2t},e^{ t} \rangle+\langle t^2,t^3,t \rangle\cdot\langle 3e^{3t},2e^{2t},e^{ t} \rangle\\
=(2t+3t^2)e^{3t}+(3t^2+2t^3)e^{2t}+(1+t)e^{t}.$$
