List<id> lstAssetIds= new List<id>();
lstAssetIds.Add('aaaaaaaaaaaaaaaaa');
List<Asset_DMSFolder__c> AssetFolders = [select Id,DMS_Folder__r.Name,Asset__r.Name,Asset__r.Id from Asset_DMSFolder__c where Asset__c IN: lstAssetIds];

AssetFolders[0].DMS_Folder__r.Name = 'renameusingchild';
update AssetFolders;
    // the above update does not change the folder name to renameusingchild
System.debug([select Id,DMS_Folder__r.Name,Asset__r.Name,Asset__r.Id from Asset_DMSFolder__c where Asset__c IN: lstAssetIds][0].DMS_Folder__r.Name);

ContentFolder cf = [select Id from ContentFolder where Id =:AssetFolders[0].DMS_Folder__c ];
cf.Name =  'testdirectobjupdate';
update cf;
System.debug([select Id,DMS_Folder__r.Name,Asset__r.Name,Asset__r.Id from Asset_DMSFolder__c where Asset__c IN: lstAssetIds][0].DMS_Folder__r.Name);
//the above update works