This isn't from a performance of the show, but it's Sara Bareilles herself performing one of the songs, an extremely nice number called, "She Used to Be Mine." And the video is well-produced, not only superimposing the lyrics, but also interspersing overlapped photographs and brief clips of the production throughout the piece.
I love the movie, for a variety of reasons that I noted in the above-linked article. And if this song is an indication of the score -- a tricky endeavor, since the film is quirky and charming and even edgy at times, while also being funny and romantic -- then they've done well.
One note about the lyrics here, so that a reference makes sense: the shy, adorable, but often put-upon title character is at a crossroads in her life and pregnant, singing about herself here, and she has one particularly-notable talent: she is a master of baking pies.
And as a bonus, here is some of the production, in a nice little behind-the-scenes video.