Identities of derived varieties

Let $A$ be a non-associative algebra with a derivation $d$. Derived operations are new "multiplications" $(x>y)$ and $(x<y)$ defined by the rule $x>y=d(x)y$, $x<y=xd(y)$. We will study the relations between the identities that hold for the derived operations and Manin products for operads. We will also consider close problems related to Rota–Baxter and averaging operators.