 Graded Leibniz rule

On a graded algebra, a left derivation of degree $r$ satisfies the displayed rule for homogeneous $a$. A right derivation of degree $r$ satisfies $R(ab)=aR(b)+(-1{)}^{r|b|}R(a)b$. For odd $r$, these reduce to the familiar parity sign rules. The left Gerstenhaber bracket makes $[u,-]$ a left derivation of degree $|u|-1$ on Hochschild cohomology; this rule holds for the induced cohomology operations, not necessarily for their cochain representatives.