I'm having some trouble with some math on a problem for a physics class (looking for help with some partial derivatives, not an answer).
Let $$L'=L+\dfrac{dF}{dt},$$ where $L$ is a Lagrangian and $F$ is a function of $q_i$ and $t$.
Show that $L'$ satisfies the Lagrange equation.
First, I expanded: $$\dfrac{dF}{dt}=\dot{q}_i\dfrac{\partial F}{\partial q_i}+\dfrac{\partial F}{\partial t}.$$
Then, I took the partial of $L'$ with respect to $\dot{q}_i$ and got that $$\dfrac{\partial L'}{\partial \dot{q}_i}=\dfrac{\partial L}{\partial \dot{q}_i}+\dfrac{\partial F}{\partial q_i},$$ but now I have to take the full derivative of this expression with respect to time, and I don't know how to evaluate $$\dfrac{d}{dt}\dfrac{\partial F}{\partial t}.$$
I started to think that I didn't need to evaluate $$\dfrac{d}{dt}\dfrac{\partial F}{\partial t}$$ because it might cancel with a term from the other part of the Lagrange equation (i.e. $\dfrac{\partial L'}{\partial q_i}$), but I got that term to be $$\dfrac{\partial L'}{\partial q_i}=\dfrac{\partial L}{\partial q_i}+\dfrac{\partial^2 F}{\partial q_i^2}+\dfrac{\partial}{\partial q_i}\dfrac{\partial F}{\partial t}.$$ Obviously all the partials of L don't matter because, since L is a Lagrangian, all those terms will cancel out, but if someone could show me what I'm missing with this function $F$, that would be greatly appreciated!
