Here is the formula I am trying to derive: $$\tau = \omega\times I \omega + I\dot{\omega}$$ where $\tau$ is the torque. $\dot{\omega}$ is the angular acceleration. $I$ is the moment of inertia. $\omega$ is the angular velocity.
This is what I tried. I know that the angular momentum:
$$L = I\omega $$
So
$$\tau = \frac{dL}{dt} = \frac{d(I\omega)}{dt} = \frac{dI}{dt}\omega + \frac{d\omega}{dt}I $$
$$\tau = \frac{dI}{dt}\omega + \dot{\omega}I $$
It is close, but I can not go further. I only know that $I = m r^2$, and $\frac{dI}{dt}$ does not give me the result I wanted. I must have missed something.