Clarifications on Backup

Is this normal?

I have monthly Normal(fullback), weekly Incremental and
daily Differential scheduled on a W2K server. However, I
observed that when a "directory" is renamed after a
fullback, only the renamed directory is backuped at the
next Incremental or Differential without the contents.
Thus the backup did not treat the whole directory with its
contents as new, but rather the new name given to the
directory.

Any way to resolve this?

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