$$F_{net} = \dfrac{d\vec{p}}{dt}$$

$$\vec{r} \times \vec{F}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$

$$\vec{\tau}_{net} = \vec{r} \times \dfrac{d\vec{p}}{dt}$$

$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \dfrac{\vec{r}}{dt} \times \vec{p}$$

$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - \vec{v} \times \vec{p}$$

$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right) - m \underbrace{\vec{v} \times \vec{v}}_{=0}$$

$$\vec{\tau}_{net} = \dfrac{d}{dt}\left(\vec{r} \times \vec{p}\right)$$

$$\vec{\tau}_{net} = \dfrac{d\vec{L}}{dt}$$