Frechet/functional derivative - misunderstanding

desmal

Hi people,

I have some confustion in understanding the frechet/functional derivative.

If we have a function like this: -

F(x)= ∫x(r')G(r',r)dr' the integration over Si (2D domain) , G is a 2D green function, r is a 2D vector outside Si, r' is a 2D vector inside Si

if I want to take the derivative δF(x)/δx at some point x=xi what should I have?

According to a book I have it becomes
F(x)= ∫xi(r')G(r',r)dr'

But I don't see the reason?