Derivative and entropy: the only derivations from $C^1(RR)$ to $C(RR)$

Let $T:C^1(RR)\to C(RR)$ be an operator satisfying the derivation equation $T(f\cdot g)=(Tf)\cdot g + f \cdot (Tg),$ where $f,g\in C^1(RR)$, and some weak additional assumption. Then $T$ must be of the form $(Tf)(x) = c(x) \, f'(x) + d(x) \, f(x) \, \ln |f(x)|$ for $f \in C^1(RR), x \in RR$, where $c, d \in C(RR)$ are suitable continuous functions, with the convention $0 \ln 0 = 0$. If the domain of $T$ is assumed to be $C(RR)$, then $c=0$ and $T$ is essentially given by the entropy function $f \ln |f|$. We can also determine the solutions of the generalized derivation equation $T(f\cdot g)=(Tf)\cdot (A_1g) + (A_2f) \cdot (Tg), $ where $f,g\in C^1(RR)$, for operators $T:C^1(RR)\to C(RR)$ and $A_1, A_2:C(RR)\to C(RR)$ fulfilling some weak additional properties.