Let $F$ be smooth function on manifold and $v$ vector field. Set
$$G(x) := dF_x(v(x)).$$
If $w$ is another vector field, how do I work out $dG_x(w(x))$? I guess should be
$$dG_x(w(x)) = d^2F_x(v(x),w(x)),$$
but am not sure how to make formalism work (how define $d^2F$?)
If you think about $G$ as a composition of maps between manifolds $$ M \mathop{\to}_v TM \mathop{\to}_{dF} \Bbb R,$$ then you can just apply the chain rule to get $D_xG(w) = D_vdF(D_x v(w))$ where the derivatives are taken in higher tangent bundles:
$$ TM \mathop{\to}_{Dv} TTM \mathop{\to}_{DdF} \Bbb R.$$
In coordinates you can expand this using partial derivatives to $DG(w) = w^i v^j F_{,ij} + w^i F_{,j} v^j_{,i}$; so if you have a Riemannian structure you get a coordinate-free formula something like what you were looking for:
$$ DG(w) = \nabla^2 F(v,w) + dF( \nabla_w v ).$$
Here $\nabla$ is the Riemannian connection and $\nabla^2$ the covariant Hessian.
