Faster Rails: Is Your Database Properly Indexed?

My Rails app used to be fast and snappy, and everything was working just fine for several months. Then, slowly, as my product grew and users started to flock in, web requests become slow and my database’s CPU usage started hitting the roof. I hadn’t changed anything, why was my app getting slower?

Is there any cure for the issues I’m having with my application, or is Rails simply not able to scale?