= Solution
Set $K=1/(4\pi\epsilon_0)$, $t_0=t-r/c$ and $\mathbf n=\mathbf r/r$, and evaluate $\mathbf p$, $\mathbf p'$, $\mathbf p''$ at $t_0$. The <chain rule> gives
$$
\nabla t_0=-\frac{\mathbf n}c,\qquad
\partial_i p_j=-\frac{n_i}c p'_j,\qquad
\partial_i n_j=\frac{\delta_{ij}-n_in_j}r.
$$
In particular, for $\mathbf u(t_0)$ and any positive integer $m$,
$$
\nabla\left(\frac{\mathbf n\cdot\mathbf u}{r^m}\right)
=\frac{\mathbf u-(m+1)\mathbf n(\mathbf n\cdot\mathbf u)}{r^{m+1}}
-\frac{\mathbf n(\mathbf n\cdot\mathbf u')}{cr^m}.
$$
Apply this <gradient> identity to the <electric potential> $\phi=K[(\mathbf n\cdot\mathbf p)/r^2+(\mathbf n\cdot\mathbf p')/(cr)]$:
$$
\nabla\phi=K\left[
\frac{\mathbf p-3\mathbf n(\mathbf n\cdot\mathbf p)}{r^3}
+\frac{\mathbf p'-3\mathbf n(\mathbf n\cdot\mathbf p')}{cr^2}
-\frac{\mathbf n(\mathbf n\cdot\mathbf p'')}{c^2r}
\right].
$$
The <magnetic vector potential> is $\mathbf A=K\mathbf p'/(c^2r)$, with time <derivative> $\partial_t\mathbf A=K\mathbf p''/(c^2r)$. Thus the <electric field> $\mathbf E=-\nabla\phi-\partial_t\mathbf A$ is
$$
\boxed{\mathbf E=\frac1{4\pi\epsilon_0}\left[
\frac{3\mathbf n(\mathbf n\cdot\mathbf p)-\mathbf p}{r^3}
+\frac{3\mathbf n(\mathbf n\cdot\mathbf p')-\mathbf p'}{cr^2}
+\frac{\mathbf n(\mathbf n\cdot\mathbf p'')-\mathbf p''}{c^2r}
\right].}
$$
For the <magnetic field>, use the <curl> product rule and the <chain rule>:
$$
\begin{aligned}
\nabla\times\left(\frac{\mathbf p'}r\right)
&=\nabla(r^{-1})\times\mathbf p'+r^{-1}\nabla\times\mathbf p'\\
&=-\frac{\mathbf n\times\mathbf p'}{r^2}
-\frac{\mathbf n\times\mathbf p''}{cr}.
\end{aligned}
$$
Consequently the <fields of a time-dependent electric dipole> include
$$
\boxed{\mathbf B=\frac{\mu_0}{4\pi}\left(\frac{\mathbf p'}{r^2}
+\frac{\mathbf p''}{cr}\right)\times\mathbf n.}
$$
These formulas hold for $r>0$. The $r^{-3}$ <electric field> is the retarded <electric dipole field>, the $r^{-2}$ terms are induction contributions, and the $r^{-1}$ terms describe <electric dipole radiation>. Retaining the dependence of $t_0$ on position in every spatial <derivative> is what produces both the induction and radiation contributions.
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