Cocentralizing and vanishing derivations on multilinear polynomials in prime rings

Let $R$ be a prime ring of characteristic different from 2 and extended centroid $C$ and let $f(x_1,\dots,x_n)$ be a multilinear polynomial over $C$ not central-valued on $R$, while $\delta$ is a nonzero derivation of $R$. Suppose that $d$ and $g$ are derivations of $R$ such that
$$ \delta(d(f(r_1,\dots,r_n))f(r_1,\dots,r_n)-f(r_1,\dots,r_n)g(f(r_1,\dots,r_n)))=0 $$
for all $r_1,\dots,r_n\in R$. Then $d$ and $g$ are both inner derivations on $R$ and one of the following holds: (1) $d=g=0$; (2) $d=-g$ and $f(x_1,\dots,x_n)^2$ is central-valued on $R$.