It depends on what derivative work means, and finding out the specifics may mean spending time in court.
GPLv3 says 'To “modify” a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a “modified version” of the earlier work or a work “based on” the earlier work.'
If you read and study a work, and then write something new that does the same thing, that may be considered adapting or copying from (parts of) the original work.
"in a fashion requiring copyright permission" is basically the definition of "derivative" here, so the GPL is rightfully farming that question out to external law and not even trying to tell you the answer.
GPLv3 says 'To “modify” a work means to copy from or adapt all or part of the work in a fashion requiring copyright permission, other than the making of an exact copy. The resulting work is called a “modified version” of the earlier work or a work “based on” the earlier work.'
If you read and study a work, and then write something new that does the same thing, that may be considered adapting or copying from (parts of) the original work.