I am given with partial differential equation $$-(au_x)_x+cu= f \qquad u(0) = u(1) = 0$$ where $a \in H^1[0,1]$ sobolev space, $c, f \in L^2[0,1]$, $c \geq 0 $ a.e. and $a \geq p > 0$ and $u \in H^2[0,1]$.
Now for $a$ as defined above we have an operator $$A(a): H^2\cap H^1_0 \to L^2$$ defined by $$A(a)\phi = -(a\phi_x)_x +c\phi$$ Now Clearly $A(a)$ is a linear operator. $$A(a)u = -(au_x)_x+cu = f \implies u = A(a)^{-1}f$$ Now for frechet derivative for some operator i need to find out $$u(a+h)-u(a)= A(a+h)^{-1}f-A(a)^{-1}f$$ Further for this i need $$u(a+h) = A(a+h)^{-1}f \implies A(a+h)u(a+h) = f$$ Then $$A(a+h)u(a+h) = -((a+h)u_x(a+h))_x+cu(a+h)$$ Now $$u(a+h) = A(a+h)^{-1}[-((a+h)u_x(a+h))_x+cu(a+h)]$$ I know the answer that $$u(a+h)-u(a) = A(a)^{-1}(hu_x)_x+O(h^2)$$ where $O(h^2)$ are terms containing powers of $h$ $\geq 2$.
Now if i define an operator $$F: D(F) = \{H^1[0,1]/ a\geq p \} \to L^2[0,1] : F(a) = u(a)$$ Then i want to compute the frechet derivative of $F$ and i know $$F(a) = u(a) = A(a)^{-1}f$$ Also the frechet derivative is $$F'(a)h = A(a)^{-1}(hu_x(a))_x$$ Then i started with $$F(a+h)-F(a) = u(a+h)-u(a)$$ Please help further.