Certain positive linear operators

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