Derive the Biot-Savart law

$$\mathbf{B}(\mathbf{r})=\frac{\mu_{0}}{4 \pi} \int_{V} \frac{\mathbf{j}\left(\mathbf{r}^{\prime}\right) \times\left(\mathbf{r}-\mathbf{r}^{\prime}\right)}{\left|\mathbf{r}-\mathbf{r}^{\prime}\right|^{3}} \mathrm{~d} V$$

from Maxwell's equations, where the time-independent current $\mathbf{j}(\mathbf{r})$ vanishes outside $V$. [You may assume that the vector potential can be chosen to be divergence-free.]