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.

-
Oh, I was informed that the accept button is mainly for the stackoverflow people, who have to test their code to see what works. Anyhow, I shall go ahead and start using it. –  Joebevo Jul 27 '12 at 7:30

$q$, $\dot{q}$ 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: $$\frac {\partial {\ddot {q} }} {\partial \dot q} = \frac {\frac {\partial {\dot q}} {\partial t}} { \partial {\dot q} } = \frac {\partial( \frac {\partial {\dot q}} {\partial q}) }{\partial t} = \frac {\partial 1} {\partial t} = 0$$