I have recently stumbled upon the notion of a coderivation. I had been working with twisted coderivations for a while, without realizing they there was a name for them (and probably some nice treatment in the literature). For me it was just an extra condition.
Without further ado, a twisted coderivation, say a $a$-coderivation (for some element $a$ in a algebra Hopf algebra $R$) is a map $d \colon R \rightarrow R$ such that
$$\Delta d(r) = d(r_1) \otimes r_2 + a r_1 \otimes d(r_2)$$
for any $r \in R$ in Sweedler notation or simply, $\Delta d = (d \otimes 1 + a \otimes d) \Delta$.
I want to find where the concept of coderivation (twisted and/or not) was first defined. Any other good references, where there is a treatment of the topic, are greatly appreciated. I have looked myself some books on Hopf Algebras like Sweedler, Montgomery and Radford. Only Radford had the definition of coderivation and some results (exercises actually) about it, but it is only a partial answer.