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The approximation of convex functions by some other convex ones with additional properties plays an important role in the theory of growth of holomorphic functions. As a rule, it is used infinitely differentiable or even real-analytic approximations. The existence of such approximations has been investigated by G. Braichev, D. Azarga, E. Abakumov and E. Doubtsov. However, they have some restrictions on the degree of proximity and cannot guarantee simultaneously the asymptotic proximity of the derivatives of the initial and constructed functions. At the same time, studying classical operators in weighted holomorphic function spaces it is necessary to have an opportunity to ap-proximate simultaneously and with an arbitrary proximity a convex function and its derivative by some convex func-tion and its derivative. It is clear, that it is possible with the help of only piecewise linear approximations. G. Braichev has previously proposed a method to construct those approximations that guarantee infinitesimal proximity of func-tions and their derivatives. In the paper it is developed a modification of this method which allow us to get desired piecewise linear approximations with an arbitrary degree of the proximity for convex functions and their derivatives on the negative real semiaxis. These convex functions form the class of radial weights defining Bergman spaces of holomorphic functions in the unit disc with sup-norms.