Can we derive Biot Savart law from Coulomb's law by setting a current source at an instant in place of one of the charges? Let's say $dl$ is the length of the conductor having $dQ$ charge which applies some force at the other charge at some instant of time. Taking derivates on both sides we get some current term. But how to prove it?

By Coulomb's law

$dE = k \frac{dq}{r^2}$

$\dfrac{dE}{dt} = k\dfrac{I}{r^2}$

$d\vec{l}\times \dfrac{d\vec{E}}{dt} = k I d\vec{l} \times \dfrac{ \vec{r}}{r^3}$

Is it correct? If yes, how should I proceed ahead?