From OS X 10.10 Yosemite, you can now use the Mac application “QuickTime Player” to record the operations of your iPhone or iPad as a captured video.

Capture iPhone/iPad video - QuickTime Player

  • Here, I tried connecting an iPhone. When you operate the iPhone, the screen of QuickTime Player will change like this, and it will be recorded properly.

Capture video/images for Apple TV (4th generation) - Mac - QuickTime Player