Product rule for partial derivatives I am going through the solution for a problem (1.7 from Goldstein's Classical Mechanics) where it says:



I don't understand why the right-hand side of the second line only contains 4 terms when there should be 5. The very last term on line 1 has been expanded into 1 term on line 2 using the product rule, but according to the product rule there should be 2 terms.
$q$, $\dot{q}$ and $\ddot{q}$ are being treated here as separate variables, so 
$\dfrac{\partial \ddot{q}}{\partial \dot{q}} = 0$.
I think this results from:
$$ {\frac {\partial {\dot q}} {\partial {\dot q}}} = 1
$$
Hence:
$$
\frac {\partial {\ddot {q}  }} {\partial \dot q}  =
\frac {\frac {\partial {\dot q}} {\partial t}} { \partial {\dot q}  } = \frac {\partial( \frac {\partial {\dot q}} {\partial q}) }{\partial t} = \frac {\partial 1} {\partial t} = 0
$$