= Solution
Let $\{\mathbf e_i(t)\}$ be an orthonormal <basis> fixed in $S'$ and write $\mathbf a=a_i\mathbf e_i$. A basis vector rotating with <angular velocity> $\boldsymbol\omega$ satisfies
$$
\left(\dot{\mathbf e}_i\right)_S=\boldsymbol\omega\times\mathbf e_i.
$$
Differentiating the components and the basis vectors in $S$ therefore gives the <rotating-frame derivative formula>
$$
\boxed{(\dot{\mathbf a})_S
=\dot a_i\mathbf e_i+a_i\boldsymbol\omega\times\mathbf e_i
=(\dot{\mathbf a})_{S'}+\boldsymbol\omega\times\mathbf a.}
$$
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