Swim up to the dolphin until it notices you. If it squeaks, you're in business, if you see red sparks from it's blowhole and it screams, you're in trouble. Keep hard element in your final hotbar slot. Dolphins aggro every other dolphin when fed. Make sure the one you want is alone. Always keep the dolphin between you and the direction you want it to go, because it will run away occasionally and you don't want it running into a pack of other dolphins or a wyrm. Sometimes they swim away after 3 pets, 2 pets and a feed, or just one pet. The bite during taming doesn't hurt much so just stay away from the mouth. The bite during true aggro hurts a lot.

Biggest tip: when the dolphin swims away STAY RIGHT ON IT because the taming bar will absolutely plummet 5-10% every second if you don't pet it fast enough. Because of this bring way more element than you'd expect to use. Around 3 stacks of 100 each should be enough. You never know when you'll have to reposition your target or avoid hostiles so be prepared to lose some progress.

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