$$ (1) \Big(\frac{d}{dt}\Big)_{rot_1} r(t) = \Big(\frac{d}{dt}\Big)_{in}r(t) - \Omega_1(t) \times r(t) $$ where $rot_1$ denotes a coordinate system rotating at $\Omega_1(t)$ relative to the fixed system $in$. $$ (2) \Big(\frac{d}{dt}\Big)_{rot_2} r(t) = \Big(\frac{d}{dt}\Big)_{in}r(t) - \Omega_2(t) \times r(t) $$ where $rot_2$ denotes a coordinate system rotating at $\Omega_2(t)$ relative to the fixed system $in$.

From (1) and (2) $$\Big(\frac{d}{dt}\Big)_{rot_1} r(t) + \Omega_1(t) \times r(t) = \Big(\frac{d}{dt}\Big)_{rot_2} r(t) + \Omega_2(t) \times r(t)$$

$$ (1) \Big(\frac{d}{dt}\Big)_{rot_1} r(t) = \Big(\frac{d}{dt}\Big)_{in}r(t) - \Omega_1(t) \times r(t) $$ where $rot_1$ denotes a coordinate system rotating at $\Omega_1(t)$ relative to the fixed system $in$. $$ (2) \Big(\frac{d}{dt}\Big)_{rot_2} r(t) = \Big(\frac{d}{dt}\Big)_{in}r(t) - \Omega_2(t) \times r(t) $$ where $rot_2$ denotes a coordinate system rotating at $\Omega_2(t)$ relative to the fixed system $in$.

From (1) and (2) $$\Big(\frac{d}{dt}\Big)_{rot_1} r(t) + \Omega_1(t) \times r(t) = \Big(\frac{d}{dt}\Big)_{rot_2} r(t) + \Omega_2(t) \times r(t)$$

Finally, $$ \Big(\frac{d}{dt}\Big)_{rot_2} r(t) = \Big(\frac{d}{dt}\Big)_{rot_1} r(t) + (\Omega_1(t) - \Omega_2(t)) \times r(t)$$