Rigidity and stability of the Leibniz and the chain rule

We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T₁, T₂,A: C^k(ℝ) → C(ℝ) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ C^k(ℝ), namely, V (f · g) = (T₁f) · g + f · (T₂g) for k = 1, V (f · g) = (T₁f) · g + f · (T₂g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T₁f) ○ g · (T₂g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T₁f) · (Ag) + (Af) · (T₂g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T₁ and T₂ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · Tg + B(f, g), and show under suitable conditions on B in the first case that B = 0 and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation.