Chapter 14 - Calculus of Vector-Valued Functions - 14.2 Calculus of Vector-Valued Functions - Exercises - Page 720: 13

$$ r'(t)=\langle 1,2t,3t^2 \rangle,$$ $$ r''(t)=\langle 0,2,6t \rangle .$$

Since $ r(t)=\langle t,t^2,t^3 \rangle $, then the derivative $ r'(t)$ is given by $$ r'(t)=\langle 1,2t,3t^2 \rangle,$$ and hence the second derivative $ r''(t)$ is given by $$ r''(t)=\langle 0,2,6t \rangle.$$