I set a Divine Wrath Card down and waited for a monster worth using the card on. He later fusion summoned a Buster Blader and made my Dragon turn to defense position. I tried using Divine Wrath against it but it didn't give me the option to activate it. Is this supposed to happen?
Divine wrath only works when the character says “My monster’s effect activates!” or something like that.
Divine Wrath can only be activated in response to a activated effect. The Dragon Destroyer's effect is a continuous effect, it doesn't activate, it just applies as soon as monster goes face-up on the field. So Divine Wrath can't be activated in this situation.
Yes - Divine Wrath can only be activated in response to a trigger-effect. Divine Wrath can't be used against permanent effects since they don't activate/trigger.