electromagnetism - Derivation of the Biot-Savart Law

Can someone provide a derivation of the Biot-Savart law for electromagnetic induction? To be clear,

$$d\vec{B}~=~\frac{\mu_0}{4\pi}\frac{I d\vec{\ell}\times \vec{r}}{r^3}.$$

Is there a simple way to compute the magnetic field at a point between two Helmholtz coils, if the radii of the coils are the same and the current through each coil is the same?