If we abide in Christ, we will produce fruit - John 15:5
I am the vine, ye are the branches: He that abideth in me, and I in him, the same bringeth forth much fruit: for without me ye can do nothing.
Christ Himself is the Vine and believers are the branches. It is not a question of the branch living its life for the vine, but simply letting the life of the vine flow out of the branches. Without Christ, we are incapable of doing anything. A vine branch has only one purpose – to bear fruit, as it is not good for anything else.