Follow-up question on "Adding a total time derivative term to the Lagrangian"

You can prove it in the following way $$dF=dF(q,t)=frac{partial F}{partial q}dq+frac{partial F}{partial t}dt$$ $$frac{dF}{dt}=dot{F}=frac{partial F}{partial q}dot{q}+frac{partial F}{partial t}$$ $$frac{partial dot{F}}{partial dot{q}}=frac{partial F}{partial q}$$

Cheers!

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On OP's comment from the second last to last equation.

$$dot{F}=frac{partial F}{partial q}dot{q}+frac{partial F}{partial t}$$ The only way for $dot{q}$ to enter in this equation is through the first term on the right, the factor $partial F/partial q$ and $partial F/partial t $ are a function of $(q,t)$. On differentiating with respect to $dot{q}$, We get

$$frac{partial dot{F}}{partial dot{q}}=frac{partial F}{partial q}frac{partial}{partial dot{q}}dot{q}=frac{partial F}{partial q}$$