Delta functions

These delta functions are starting to scare me. They are so creepy.

p18

source: QFT for the gifted noob

Attempt:

For the first part (1.44)

Write the first phi function as a trivial functional.
Use the functional derivative definition.
Evaluate one of the delta functions, dropping the integral and replacing the dummy variable with either x or y.

For the second part (1.45)

Do the same thing, but then get stuck near the end because it's more complicated.
Here the phi dot is a derivative, that we can rewrite in terms of phi using the definition of a derivative.
Then we can introduce variation and simplify terms, but I can't get a d/dt out of the mess.
So I'm thinking somehow, when I do the h-> 0 limit of the definition of phi derivative, somehow I can get a d/dt out of it?

If instead, I treat the variation as epsilon * d '(t' -t), I can't integrate by parts to get rid of the delta function derivative, since the other function is the delta function and I would have to take the derivative of that... then I'd get another delta function derivative...