This is my first build... But jax is my main and ive researched ALOT of builds with him and found this one is the best one... The reason i go withthe PD which many people dont agree with is the fact that at lvl 6 you start to jungle and it helps immensely as normally you dont get fed enough to buy a rageblade by lvl 6. exhaust and ignite is generally the best spells you can have as you are a up in the face summoner
as for the IE i choose that for the simple fact that it increases your damage out put immensely whilst having a higher crit chance as you already have the speed and Lifesteal you need.
You can choose to swap the IE with a deathcap or a Black cleaver

ok item order should go like this for maximum ability to jungle with ease.. which means quicker lvling and more PWNAGE (which we all want).

Dorans shield (excellent for first blood as it has hp and armor so you can take more hits)
Ninja Tabi (for obivous reason. His e is a excellent way to own people and with your skill + ninja tabi + runes you evade alot of attacks which makes your e skill proc unlimited in 1v1's)
Vampiric Scepter (start your lifesteal early for more survivability and as you will need it for lvl 6 when you start jungling)
Dagger (for when you hit lvl 6 and start jungling. it makes it alot easier and faster to jungle which faster gold and leveling)
Zeal (gives you crit strike but more importantly Early game, combined with your nimbleness it allows you to easily chase down anyone and stun them for a kill)
Rageblade ( A MUST for jax... gives you everything you need. AP, which also gives you hp, and attack speed to get to double stacks in no time at all... DO NOT SKIP ON THIS ITEM)

At this point you should be dominatng and ganking all over the place whilst sweeping through your jungle and helping lanes with ease. Then you start the big guns (pardon the pun)

3x Gunblades (these are awesome for jax as you gain Lifesteal, and spell vamp which is almost unbeatable. It also give jax more hp through the 80AP per gunblade you gain. I choose to go with the cutlass first as it increases damage output and is more effective at lower lvls)

At this point you've generally A: Won the game already B: they have surrendered or C: they are just letting you get more kills
If it C then you should be gettig your Last item.

Infinity Edge (this is always debated when i play jax as people say that jax doesnt need crit.. BOLLUX... crit makes you rip through your opponents with ease and the extra damage form it puts your crit at about 600.. you do the maths)