The chain rule as a functional equation

We consider operators T   from C1(R)C1(R) to C(R)C(R) satisfying the “chain rule”T(f∘g)=(Tf)∘g⋅Tg,f,g∈C1(R), and study under which conditions this functional equation admits only the derivative or its powers as solutions. We also consider T   operating on other domains like Ck(R)Ck(R) for k∈N0k∈N0 or k=∞k=∞ and study the more general equation T(f∘g)=(Tf)∘g⋅AgT(f∘g)=(Tf)∘g⋅Ag, f,g∈C1(R)f,g∈C1(R) where both T and A   map C1(R)C1(R) to C(R)C(R).