= Solution
Write the position in the rotating basis as $\boldsymbol r=\sum_i x_i'\boldsymbol e_i'$. Differentiating with respect to time in the inertial frame,
$$
\frac{d\boldsymbol r}{dt}
=\sum_i\dot x_i'\boldsymbol e_i'
+\sum_i x_i'\frac{d\boldsymbol e_i'}{dt}
=\boldsymbol v'
+\boldsymbol\omega\times\sum_i x_i'\boldsymbol e_i'.
$$
Here $\boldsymbol v'=\sum_i\dot x_i'\boldsymbol e_i'$ is the relative <velocity> measured in the <rotating reference frame>. Consequently
$$
\boxed{\boldsymbol v=\boldsymbol v'+\boldsymbol\omega\times\boldsymbol r.}
$$
This is the vector differentiation rule for a <rotating reference frame> with a common origin: the extra term accounts for the changing basis, not a translational motion of that origin.
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