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 $V,T_1,T_2,A\colon C^k(\mathbb R)\to C(\mathbb R)$ satisfy equations of the generalized Leibniz and chain rule type for $f,g\in C^k(\mathbb R)$, namely, $V(f\cdot g)=(T_1f)\cdot g+f\cdot(T_2g)$ for $k=1$, $V(f\cdot g)=(T_1f)\cdot g+f\cdot(T_2g)+(Af)\cdot(Ag)$ for $k=2$, and $V(f\circ g)=(T_1f)\circ g\cdot(T_2g)$ for $k=1$. Moreover, for multiplicative maps $A$, we consider a more general version of the first equation, $V(f\cdot g)=(T_1f)\cdot(Ag)+(Af)\cdot(T_2g)$ for $k=1$. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators $V$, $T_1$ and $T_2$ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, $T(f\circ g)=Tf\circ g\cdot Tg+B(f\circ g,g)$ and $T(f\cdot g)=Tf\cdot g+f\cdot Tg+B(f,g)$, 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.