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Derivation of the Angular Momentum Principle
$$F_{net} = \dfrac{d\vec{p}}{dt}$$
$$\vec{r} \times \vec{F}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$
$$\vec{\tau}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \dfrac{d\vec{r}}{dt} \times \vec{p}$$
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \vec{v} \times \vec{p}$$
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - m \underbrace{\vec{v} \times \vec{v}}_{=0}$$
$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right)$$
$$\vec{\tau}_{net} = \dfrac{d\vec{L}}{dt}$$