I'm looking for properties of the *Newton-derivative*, defined as follows: A function $F \colon X \to Y$ is Newton differentiable at $x\in X$ if there exists $\varepsilon>0$ and a function $G\colon B_\varepsilon \to L(X,Y)$ such that
\begin{align*}
\lim_{y\to0} \frac{\|F(x + y) -F(x) -G(v+y)[y]\|_Y}{\|y\|_X} = 0.
\end{align*}

$G$ is called Newton-derivative. Can you provide me with a proof (or any reference) for a chain rule for this differential?

Actually, I need to know about the Newton-derivative of the function 
\begin{align*}
L^2(\Omega) \ni u \mapsto \max(-1,\min(1,u)), 
\end{align*}
which hopefully exists... Thanks a lot, Malte.