Consider a simple nonlinear operator $F:W^{2,2}(0,1)\to L^2(0,1)$ where $F: w \mapsto w^3$. Assume $F$ is Frechet differntiable with derivative $DF_{w}h=3w^2h$. To prove it, we need to show: $$\frac{\|(w+h)^3-w^3-3w^2h\|_2}{\|h\|_{2,2}}\to0$$
This boils down to
$$\frac{\|h^3+3h^2w\|_2}{\|h\|_{2,2}}\to 0$$
Sobolev embedding theorem of $W^{2,2}\subset \subset L^6 $ tells us that $$\frac{\|h^3\|_2}{\|h\|_{2,2}}\to 0$$ But the last part still remains to prove; that is; showing that $$\frac{\|3h^2 w\|_2}{\|h\|_{2,2}}\to 0 \quad \forall w\in W^{2,2}$$ Please comment on this!
You can use the same embedding $W^{2,2}\hookrightarrow L^6$: $$ \|h^2w \|_{L^2} \le \|h\|_{L^6}^2 \|w\|_{L^6} \le \|h\|_{W^{2,2}}^2\|w\|_{W^{2,2}}. $$