= Solution
Put $\mathbf W=\nabla\times\mathbf F$ and $\mathbf v=\dot{\mathbf r}$. All <vector fields> and their spatial <partial derivatives> below are evaluated at $\mathbf r(t)$. The <product rule> for a <dot product> gives
$$
\frac d{dt}\bigl(\mathbf W(\mathbf r(t))\cdot\dot{\mathbf r}(t)\bigr)
=\frac{d\mathbf W(\mathbf r(t))}{dt}\cdot\dot{\mathbf r}(t)
+\mathbf W(\mathbf r(t))\cdot\ddot{\mathbf r}(t).
$$
Because the <vector field> has no explicit time dependence, the <multivariable chain rule> gives $d\mathbf W/dt=J_{\mathbf W}\dot{\mathbf r}$, where the <Jacobian matrix> has entries $(J_{\mathbf W})_{ij}=\partial_jW_i$. Consequently
$$
\boxed{\frac d{dt}\left[(\nabla\times\mathbf F)\cdot\dot{\mathbf r}\right]
=\dot{\mathbf r}^{\,T}J_{\nabla\times\mathbf F}\dot{\mathbf r}
+(\nabla\times\mathbf F)\cdot\ddot{\mathbf r}}.
$$
The first term measures the change of the <curl> sampled as the point moves; the second measures the change in the <velocity vector> used in the <dot product>.
To give the result explicitly in $P,Q,R$, define the <directional derivative> operator $L=\dot x\partial_x+\dot y\partial_y+\dot z\partial_z$. Since
$$
\mathbf W=(R_y-Q_z,\ P_z-R_x,\ Q_x-P_y)^T,
$$
the same expression is
$$
\begin{aligned}
&\dot x\,L(R_y-Q_z)+\dot y\,L(P_z-R_x)+\dot z\,L(Q_x-P_y)\\
&\quad+\ddot x(R_y-Q_z)+\ddot y(P_z-R_x)+\ddot z(Q_x-P_y).
\end{aligned}
$$
For example $L(R_y-Q_z)=\dot x(R_{yx}-Q_{zx})+\dot y(R_{yy}-Q_{zy})+\dot z(R_{yz}-Q_{zz})$; the other two terms expand in the same way. Thus only the first and second spatial <partial derivatives> of $P,Q,R$, together with the <velocity> and <acceleration> of the <parametric curve>, occur.
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