The chain rule as a functional equation (Q600970)

Motivated by the classical chain rule of the derivative \(D: C^1 (\mathbb R) \to C(\mathbb R)\), \(D(f \circ g) = ((Df)\circ g)\cdot Dg\), the authors study the general functional equation \(T(f \circ g) = (Tf) \circ g \cdot Tg\) for various classes of operators \(T\) from \(C^1(\mathbb R) \) into \(C(\mathbb R)\), or in other domains like \(C^k (\mathbb R)\). Some generalized forms are also studied.