This is a comment but just too long of one to go in comments. Both other answers are kind of right. MASTERMIND_ED has the rule cited but you can look at the rule for 20 seconds and figure out that it is missing a lot of clarity.
The rules committee will certainly address the shortcomings of this rule in the offseason. If you watched the Packers game last night (11/20/16 vs Redskins) you will note that Rodgers went head first twice and both times he was declared to have given himself up. One of the times he got up and tried to run more... what a mess the NFL has gotten themselves into.
Not only is there a large amount of ambiguity on what it means to give oneself up, but there is a no frills circumvention to this rule - slide head first. Nobody can hit you because maybe you did give yourself up and then if they don't hit you, you can get up and start running again. Completely ridiculous. And this is only an NFL problem, as other leagues the knee down is a down play. To make matters worse 80-90% of the slides are by QBs.
So the correct answer at this point in time is that there is a rule that clearly states that a player is down at the beginning of their slide when going feet first (the NFL hasn't spotted this right all year and gives QBs 1-2 yards extra per slide). Letter d - addressing feet first sliding - isn't even how it is called in games. The rule says "the instant a runner touches the ground". That is not how NFL referees call it. The runner is down as soon as he starts his slide, which affords him extra protection from being hit. This is unarguable since there are 20 late hits a year that happen far from the runner beginning his slide but close to when the runners body hits the ground.
Now the other side of this is when the QB slides head first. This wasn't meant to be a dead ball, however to keep up with player safety and the horribly vulnerable position a player is in (head facing the defense, laying on the ground), NFL referees have in almost all cases this year declared the player as giving himself up when the QB slides head first (I simply can't remember a non-QB sliding head first that wasn't touched - please comment if there is one).
So Rodgers being the smart QB he is tried to use a loophole to gain protection and then get back up. The referees were smarter than Rodgers in both cases and declared that he declared himself down. To me this is rather interesting because unlike the feet first slide, and to go along with the "down or not" ambiguity, we also have an issue with the spot of the ball.
On a face first slide you are moving forward. If you are talking about the inches of the game and not safety you certainly would want your defensive player cremating Rodgers to physically knock his body backwards. All coaches know that if you pat a sliding player on the back, the referees are more apt to give him a few extra inches (feet) after the touch. If you stop him in his tracks, you might help your cause by a few inches. Then you have the QB with the ability to possibly get up because the rules don't clearly say he is down.
This basically makes the referees make a both strategic and safety decision on every head first slide. I think the NFL will continue to rule to QB down on any slide and amend the rules ASAP. As a coach I could certainly see telling your players... Just hover a couple yards from him after a slide and lay him out if he gets up without a whistle.