Properties (including the approximating ones) are investigated of positive linear operators Ln(f; x) for which the relation $$L_n \left( {\left( {t - x} \right)f\left( t \right); x} \right) = \frac{{\varphi \left( x \right)}}{n}L'_n \left( {f\left( t \right); x} \right)$$ is fulfilled, as well as the properties of operators Ln (m)(f;x). The results are applicable, in particular, to Bernstein polynomials, to the operators of Mirak'yan, Baskakov, and others.