The main purpose of this paper is to prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L(X) be the algebra of all bounded linear operators on X and let A(X) ⊆ L(X) be a standard operator algebra. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(An) = D(An−1 )A+An−1D(A)+D(A)An−1+AD(An−1 ) for all A ∈ A(X), where n ≥ 2 is some fixed integer. In this case D is of the form D(A) = [A, B] for all A ∈ A(X) and some fixed B ∈ L(X), which means that D is a linear derivation. In particular, D is continuous.