A note on operator equations describing the integral

We study operator equations generalizing the chain rule and the substitution rule for the integral and the derivative of the type fog + c=I (Tfog.Tg), f,g ∈ C¹(ℝ), (1) where T: C¹(ℝ) → C(ℝ) and where I is defined on C(ℝ). We consider suitable conditions on I and T such that (1) is well-defined and, after reformulating (1) as V(fog)=Tfog.Tg, f,g ∈ C¹(ℝ) (2) with V: C¹ (ℝ) → C(ℝ), give the general form of T, V and I. Simple initial conditions then guarantee that the derivative and the integral are the only solutions for T and I. We also consider an analogue of the Leibniz rule and study surjectivity properties there.