Left-back Hoganson, who can also play in a wide midfield role, linked up with the Rams last month and featured in the 1-0 pre-season friendly victory at Burton.
The 18-year-old left-back was a free agent after being released by Newcastle at the end of last season.
Manager Nigel Clough said: "We're pleased Michael has decided to join us; he's a young player who has the potential to improve.
"Primarily, we're looking for him to be part of the new Under-21 side, but he will also provide cover down the left-hand side of the first-team squad."
Hoganson joined Newcastle in 2008 and featured in their reserves, under the management of Peter Beardsley, but failed to break into the Magpies' first-team squad.