Oh yes, a fair warning to those who are still using an iPhone 5 / DON'T UPDATE TO IOS 9!!! I was stupid enough to do so and it has made my iPhone as slow as thick mud through a funnel.
Has it been at least 24 hours since you upgraded, and has the device sat in a charger unused during that time?
iOS major upgrades (8 to 9 here) wipe all device cache, and frequently wipe the spotlight DB. The slowness you are seeing right now may be tied to those caches and optimizations needing to be rebuilt. The phone is doing some of this in the background, and can usually complete it overnight when sitting in a charger.
iOS 9 has some changes in it specifically meant to make older devices at least stay at iOS 8 performance, and overall will improve performance and battery life a little bit.
http://arstechnica.com/apple/2015/09/ios-9-on-the-iphone-4s-a-stay-of-execution-nothing-more/
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