How to find out the frechet derivative of the following non linear function?

Question

The given operator is $$F: D(F) = \{c \in L^2[0,1] : c \geq 0 \ a.e. \} \to L^2[0,1]$$ $$F(c) = u(c)$$ where $u$ is the solution of $$-u_{xx} + cu = f, \quad f \in \ L^2[0,1]$$ Now for $c \in D(F)$ let $$A(c) : H^2 \cap H_0^1 \to L^2$$ be defined by $$A(c) \phi = -\phi_{xx}+c \ \phi$$ Then we have $$F(c) = u(c) = A(c)^{-1}f$$ Note that $H^2$ is Sobolev space i.e. $$H^2(U) = W^{2,2}(U) \quad \& \quad H_0^1 = \overline{C_c^{\infty}(U)}$$ i.e. closure of the space of infinitely differentiable functions with compact support in $U$. Closure is taken in $W^{1,2}(U)$. Now my problem is to find the first frechet derivative of $F$. This is given by $$F'(c)h = -A(c)^{-1}(hu(c))$$ But I am unable to find it. I start with definition but strucked in the beginning.