There's something I don't get with the divergence in the conservation law $$ \frac{\partial \rho}{\partial t} + \nabla \cdot f(\rho) = 0 $$ with $$f:\mathbb{R}\mapsto \mathbb{R}^n$$ the flux. We can write, using the chain rule, $\nabla \cdot f(\rho) = f'(\rho) \cdot \nabla \rho$, right ?
So if we take $f(\rho) = \boldsymbol{v}\rho $ we have $$\nabla \cdot f(\rho) = \boldsymbol{v} \cdot \nabla \rho$$.
But at the same time, if we develop the divergence of $\boldsymbol{v}\rho$ and develop using the product rule, we have $$\nabla \cdot (\boldsymbol{v}\rho) = \boldsymbol{v}\cdot \nabla \rho + \rho \nabla \cdot \boldsymbol{v}$$
So what did I miss ?
The chain rule you’ve written is only valid if $f$ only depends on $\boldsymbol x$ through $\rho$, which is not the case if $\boldsymbol v$ also depends on $\boldsymbol x$.