Which is the correct way to go now? I love it when you have built full ap and then watch a fireball destroy an adc or mid laner, but I feel that damage wise you should build with attack damage and speed. What does everyone else build?
or 
This is for Shyvana right? Most go AP if they're vs squishy comps and attack speed for tankier comps!
Also most go 
rather than because the on hit damage of red smite. Depends if you think they'll fight or run when you gank them but you have an MS steroid on her so red is almost always better I'm p sure. Also slows aren't multiplicative so blue smite makes the frozen mallet passive redundant
When building AD or AP, it really depends on the stats of your champion really. For example, Ahri is an AP champion since all of her attacks do magic damage, not physical damage so it would be better to build AP on her not AD.
Same thing goes for junglers, it depends on the champion you play. If you play a champion like Amumu you should build AP, but if you play a champion like Master Yi it would be better to build AD. There are some champions you can build both though if you wanted, champions like Jax or Udyr (to my knowledge) are pretty damn good if you build either or.
on what champion
