You can differentiate each component as long as the unit vectors are not time-dependent. For fixed unit vectors i and j, it looks like your vector function is
x(t) = (t3 - 2t2 + t)i + etsin(t)j
dx/dt = (3t2 - 4t + 1)i + [etsin(t) + etcos(t)]j
Again, this is valid only if the unit vectors i and j are constant.