There is nothing to suggest that Jalen Rose and Derrick Rose are related. Jalen was born and raised in Detroit in the Bill Laimbeer era. His biological father is Jimmy Walker (who started with Jerry West). Though they corresponded, Jalen never met his father in person.
Derrick was born and raised without a father in the tough Englewood section of Chicago. He goes by the nickname “Pooh” which was giving to him by his grandmother (based on his resemblance to Winnie-the-Pooh as an infant).
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Surnames in families change more frequently than most people realize. Two people can share a surname and have no biological relationship to each other.
Common surnames, such as Smith and Jones, can have multiple independent founders. As well, adoptions, misspellings, name changes (sometimes used to hide a criminal or immigration past) and non-paternities (the husband is not the biological father of the child) all serve to confuse the picture.
See GenealogyinTime.com's Are People with the Same Family Name Related? to know more.