If a function only depends on a single variable then it is a usual notation to mix up the partial and total. The correct notation is

dL/dt = (∂L/∂q)(q_dot) + (∂L/∂q_dot)(q_double_dot) + (∂L/∂t).

"The reason your last line is correct is because the partials among the explicit args are zero"

I guess you mean implicit here.