In the episode The Doctor's Daughter, Jenny was named as The Doctor's "daughter". She was the progenation of The Doctor, a direct descendent. Is it possible then for Jenny to be able to regenerate? The show doesn't say it outright in the episode. Does it say it somewhere else? Would it be safe to assume otherwise?
Who knows? The mechanic's rules aren't explicit, but they are definitely changing.
In Old Who, regeneration was implied/outright stated (depending on your admission of novelizations to your own personal canon and your interpretation of certain lines in the show) to be a biotechnological ability, not an inherent feature of Gallifreyan biology. It was granted to Gallifreyans upon proving their worth (passing tests) and was part of becoming a Time Lord. (That Time Lords were a Gallifreyan (bio)technologically-privileged upper class was undeniably stated in the show.)
During this time, the specific mechanics of regeneration weren't made clear but based on what we saw it consisted of a complete biological revision/reshuffling; if you regenerated, you got a new body, and that was the way it was. The original concept (never stated onscreen) was that regeneration returned the Time Lord to a younger version of himself. At any rate, the biological revision was the healing process, not something that happened after it fixed your existing body.
(Also, control over the process --that is, deciding what you looked like-- was implied to be possible if you triggered regeneration deliberately but not if it was an automatic response to lethal trauma.)
In New Who, we've seen additions to the concept such as (spoilers!):
- A Time Lord continues to heal damage sustained during the post-regeneration afterglow --including the regrowth of lost body parts-- without further biological revision, not unlike a D&D 3.5 troll's regeneration ability.
- The regeneration process can be redirected into a suitable receptacle after allowing it to heal the triggering damage but before it makes any revisions to the subject's biology.
- A regeneration can affect non-Gallifreyans by proximity.
- A part-near-human clone of a Time Lord can revive from death through the use of automatic regeneration-like powers without changing her appearance --despite the lack of a suitable receptacle to shunt the energy into, which was Ten's excuse, and despite the fact that the Doctor has repeatedly said (in New and Old Who both) that sudden death is still permanent death: regeneration needs a window of opportunity to function in. (We'll let The Movie sit by itself on that one and pretend that the "anesthetic" handwave provides justification for an awesome scene.)
- Regeneration can have a physically violent effect on the immediate environment.
- A human conceived in the TARDIS while in motion through the Time Vortex can have full Gallifreyan DNA (seriously?) and this means she also has the full regeneration features including healing, body alteration, and afterglow immunity --as well as at least some control over her appearance even when the process was triggered automatically.
So... who knows what the Doctor's Daughter (whose DNA is part near-human as well as part Gallifreyan) is capable of? The rules for regeneration (both how it works and how it's acquired) were never made clear and are now being changed in ways that definitively contradict what was previously stated.
Doctor Who was never attached to a firm canon anyway, although it does delight in making continuity references. While Old Who's internal consistency was maintained --as much as it was-- largely thanks to its being content to simply wave its hands and move on, New Who is much more inclined to try explaining things, and that gets it into trouble.
Possibly. I mean though Martha said she probably couldn't, it's still a possibility. When she came back to life at the end of the episode, it wasn't really a regeneration, because she would've let out all that golden energy, and change her face (I think it was something to with the Source). After all, Time Lords can't regenerate when they're dead. This is demonstrated by both the Master and the Doctor.