Method from company rising in demand (7)
I believe the answer is:
process
'method' is the definition.
(I have seen 'Method of production' mean 'process' so perhaps 'method' could also mean 'process')
'company rising in demand' is the wordplay.
'company' becomes 'co' (abbreviation).
'rising' is a reversal indicator (going up in a down clue).
'in' is an insertion indicator.
'demand' becomes 'press' (press is a kind of demand**).
'co' reversed gives 'oc'.
'oc' inserted inside 'press' is 'PROCESS'.
'from' is the link.
(Other definitions for process that I've seen before include "Natural series of changes" , "Course of action intended to achieve a result" , "Series of actions to achieve a result" , "Move in formation" , "Series of actions with a single purpose" .)