= Solution
The <retarded potential> represents causal propagation: each source element contributes according to its charge density at the earlier time needed for a signal to reach the observer, with inverse-distance weighting. The printed formula uses units with the speed of light equal to one.
Insert a <Dirac delta function> in time to write
$$
\phi(t,x)=\frac1{4\pi\epsilon_0}\int d^3x'\int d\tau\,\frac{\rho(\tau,x')}{|x-x'|}\delta(t-\tau-|x-x'|).
$$
For the moving point charge, spatial integration leaves $q/(4\pi\epsilon_0)$ times $\int R(\tau)^{-1}\delta(t-\tau-R(\tau))\,d\tau$, with $R(\tau)=|x-x_0(\tau)|$. The retarded time $\tau_r$ solves $t-\tau_r=R(\tau_r)$. Since $R'(\tau)=-v(\tau)\cdot n(\tau)$, the delta-function Jacobian is $1-v\cdot n$, positive for subluminal motion. Therefore the <Liénard–Wiechert potential> is
$$
\boxed{\phi(t,x)=\frac q{4\pi\epsilon_0(R-v\cdot R)},}
$$
where $R=x-x_0(\tau_r)$ and $v=\dot x_0(\tau_r)$. Replacing charge density by the current density $qv\delta^{(3)}(x-x_0)$ gives $\boxed{A=v\phi}$ in these $c=1$ units. Restoring SI units gives denominator $R-v\cdot R/c$ and $A=v\phi/c^2$, with all source quantities still evaluated at retarded time.
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