= Solution
Write the geometric <vector> as $\mathbf V=\sum_iV_i'\mathbf e_i'(t)$. A rotating orthonormal basis obeys $\dot{\mathbf e}_i'=\boldsymbol\Omega\times\mathbf e_i'$. The <product rule> gives the transport identity
$$
\boxed{\left(\frac{d\mathbf V}{dt}\right)_{\rm inertial}=\left(\frac{d\mathbf V}{dt}\right)_{\rm rotating}+\boldsymbol\Omega\times\mathbf V.}
$$
Both terms must be expressed in the same basis before adding them. If $R(t)$ maps rotating components into inertial components, the precise component equation is $d(R\mathbf V')/dt=R(\dot{\mathbf V}'+\boldsymbol\Omega'\times\mathbf V')$. The printed expression suppresses this basis conversion.
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