On derivatives of superposition operators between the spaces $L_p$

In the first part of the article there are described some notions of the derivatives for nonlinear operators useful in applications. The main results about these derivatives and the comparative analysis are presented. The second part is devoted to the differentiability properties of the simplest nonlinear operator, the superposition operator, $\mathsf{f}x(s) = f(s,x(s))$ in the spaces $L_p$ ($1 \le p \le \infty$); in particular, the conditions of the differentiability in different senses (presented above) of these operators and their continuity and uniformly continuity on bounded sets.