On identities for derivative operators

Let X be a commutative algebra with unity e and let D be a derivative on X that means the Leibniz rule is satisfied: D(f\cdot g)=D(f)\cdot g+f\cdot D(g). If D^{(k)} is k-th iteration of D then we prove that the following identity holds for any positive integer k frac{1}{k!}\sum\limits_{j=0}^k(-1)^j\binom{k}{j}f^jD^{(m)}(gf^{k-j})=\Phi_{k,m}(g,f)=\begin{cases}0,\ 0\leq m As an application we prove a sharo version of Bernstein’s inequality for trigonometric polynomials.