Slow Start Up after Yosemite Update (not upgrade)

You'll need to boot up on an external bootable drive so you can repair your internal disk. Use Disk Utility from the external drive to run repair permissions and disk.

You should also free up some space on it as well here. Get rid of any old files you don't need or have backups of. There are some deeper cleaning apps you can use to delete log & cache files that could be choking your system up as well.

The very last thing here which we do yearly is defragment the hard drive (not SSD units). Over time your OS files get fragmented as each update extends them to other parts of the disk which then slows the system. But before you defrag the disk make sure to get rid of the junk first!