You set the playback speed to 60fps. That has little to do with how many actual frames are in your animation. You'll need to add additional frames if you want more frames when rendering to image.

If you're rendering to avi or mov (depending on your machine), to slow down the frame rate (giving the appearance of more frames) you can decrease the fps. To speed it up, you increase the fps.  So at 30 fps it would take 30 frames to render one second of animation. At 60 fps, it would take 60 frames to render one second of animation. The computer won't just make new frames for you though, you have to do that yourself.