The chain rule as a functional equation

We consider operators T from C1(R) to 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) for k ∈ N0 or k=∞and study the more general equation T (f ◦ g) = (Tf ) ◦ g · Ag, f, g ∈ C1(R) where both T and A map C1(R) to C(R). © 2010 Elsevier Inc. All rights reserved.