What are we looking at in your screenshot? I've been Where's-Waldoing it for the word 'destroy' but can't find it.
Command was:
apt-get -y install binutils ; dd if=/dev/vda bs=1M | strings -n 100 | grep 2013-12
The destroy api call docs are here:https://developers.digitalocean.com
(It's the /droplets/[droplet_id]/destroy one.)