$T(f\cdot g)=(Tf)\cdot g + f \cdot (Tg),$
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.
J. Aczél, "Lectures on Functional Equations and their Applications," Mathematics in Science and Engineering, 19, Academic Press, New York-London, 1966.
S. Artstein-Avidan, H. König and V. Milman, The chain rule as a functional equation, Journ. Funct. Anal., 259 (2010), 2999-3024.doi: 10.1016/j.jfa.2010.07.002.
H. König and V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule, preprint.