The Magicians' Guild (The Black Magician Trilogy #1) by Trudi Canavan

"We should expect this young woman to be more powerful than our average novice, possibly even more powerful than the average magician."
This year, like every other, the magicians of Imardin gather to purge the
city of undesirables. Cloaked in the protection of their sorcery, they
move with no fear of the vagrants and miscreants who despise them and
their work-—until one enraged girl, barely more than a child, hurls a
stone at the hated invaders...and effortlessly penetrates their magical
shield.

What the Magicians' Guild has long dreaded has finally
come to pass. There is someone outside their ranks who possesses a raw
power beyond imagining, an untrained mage who must be found and schooled
before she destroys herself and her city with a force she cannot yet
control.