Can Pranayama Be Derived from Anapana? Pranayama seam to have some similarities with Anapana. What is the relationship between the two. Was this derived from Anapana? What is it's historic development?
Yes it certainly got some similarities. Yeah buddhism originated in India and there were 27 buddhas before the Gauthama Buddha so there are some techniques remianed from others. But it certainly is the same. 
In Pranayama there is a rhythm of exhale and inhale. But in Anapansathi. Which is "Ana-(inhale)-pana-(exhale)-sathi(concentration)" This just keep concentrating on the Exhale and inhale in a little while you feel calmness you never felt before and slowing things down. 
This is the last meditation technique Buddha has done before the Enlightnement. 
There are 4 other meditation techniques to support like Mithree Meditation where we bless ourselves and all animals and everything we can see and cannots including our enimies.
This all comes under Samatha Meditation. So if you want to goto Anapanasathi which is under Vipassana Mediation. You have to follow the Samatha meditation to get a relief for body and mind. I guess you get the answer.