Is accounting and finance a direct and Productive Department or Indirect Department of any Manufacturing Firm?

Its productive department . Support provided on following issues make finance department an integral and productive department without which business can not make rational decision. 

1. Inventory management
2. working capital management
3. production losses
4. Analysis on actual cost comparing with standards covering following issue :
   1. Material 
   2. Labour
   3. Fixed Overheads
   4. Variable Overheads
5. Product costing , which help to determine pricing
6. market segment analysis , with cost effectiveness
7. concept of opportunity cost in various decisions

indirect

it mean, that we support others to do thier jobs effeciently???? 

Yes both of them are direct and productive departments because without financing you cannot do any manufacturing and as they say "accounting is the language of business" means if accounting is not there in any type of business, you won't know where you started and where you are now?.