= Solution
The <electric dipole moment> of a localized charge density is
$$
\mathbf p(t)=\int_{\mathbb R^3}\mathbf x\,\rho(\mathbf x,t)\,d^3x.
$$
Differentiate and use the <continuity equation> $\dot\rho+\nabla\cdot\mathbf J=0$:
$$
\dot p_i
=-\int x_i\partial_jJ_j\,d^3x
=\int J_i\,d^3x
-\int_{\partial\mathbb R^3}x_iJ_jn_j\,dS.
$$
Localization makes the boundary integral vanish. Component by component,
$$
\boxed{\dot{\mathbf p}(t)=\int_{\mathbb R^3}\mathbf J(\mathbf x,t)\,d^3x}.
$$
This is the <time derivative of the electric dipole moment>.
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