How do you take out a affected analog stick from the motherboard?

In order to replace a single analog stick you will need to have the ability and tools needed to unsolder and solder in a new stick. They are not meant to be individually replaced. The easiest way to replace a faulty analog stick is to replace the entire motherboard per the DualShock 4 CHU-ZCT2U Motherboard Replacement guide. If you do have the tools to solder in a new analog stick like the ElecGear analog stick replacement, if you do this it is recommended that you replace both sticks at the same time due to one failing there is potential for the other to need replacing soon.