How to do functional derivatives

[leroyjenkens]

Here's an example from my homework. I already turned it in, though. I basically just copied what I could from my notes, but I have no idea how this is done.
Could someone explain this to me? I can't find anything intelligible (at least to me) of this stuff on any website. My notes contain parts where the instructor changed certain terms into delta functions, but I'm not sure why, and I don't know when to do that.
So if anyone has the time, could you explain this step by step to me of what this question is looking for? I'll post the picture from my homework.
Thanks.

 

[micromass]

Please post in the homework section.