Consider $F(f) = f^2$ with $F:L^2[0,1] \to L^1[0,1]$. We want to know if it's differentiable by Frechet.
So based on definition:
$$ \underset{\|h\|_{L_2} \to 0}{\lim}\dfrac{\|(f+h)^2 - f^2 - A(f) h\|_{L_1}}{\|h\|_{L_2}} = \underset{\|h\|_{L_2} \to 0}{\lim} \dfrac{\|h^2 +2fh - A(f)h\|_{L_1}}{\|h\|_{L_2}} = \star $$
Choosing $A(f) = 2f$ we will have $\star = \underset{\|h\|_{L_2} \to 0}{\lim} \|h\|_{L_2} = 0$.
Is it true, or maybe I miss some part?