= Solution
The <contraction of two Levi-Civita symbols> gives $[F\times(\nabla\times G)]_i=F_j\partial_iG_j-F_j\partial_jG_i$. Similarly $[G\times(\nabla\times F)]_i=G_j\partial_iF_j-G_j\partial_jF_i$. Add the two directional-derivative terms to cancel the last terms, leaving
$$
F_j\partial_iG_j+G_j\partial_iF_j=\partial_i(F_jG_j).
$$
The <product rule> has therefore proved the <gradient of a dot product> identity
$$
\boxed{\nabla(F\cdot G)=(F\cdot\nabla)G+(G\cdot\nabla)F+F\times(\nabla\times G)+G\times(\nabla\times F).}
$$
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