I have a functional of the the following form, $(o<a<1)$ :
$F(g(x)) = \int_0^a \! g(x) x \, \mathrm{d}x. + \int_a^1 \! (g(x)-k)x^2 \, \mathrm{d}x. $
I want to find $ \frac{\partial F(g(x))}{\partial g(x)}$.
I see here that the functional derivative has linearity property : https://en.wikipedia.org/wiki/Functional_derivative#Properties. But I am still concerned because the limits of integration is different for the two parts that are summed. Will then the result be $ \frac{\partial F(g(x))}{\partial g(x)}= x + x^2$?
This does not seem right. But I am not sure how to handle this situation of different integration limits for the two parts of the functional. Any help is greatly appreciated.