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In this paper we consider the problem of counting algebraic numbers $\alpha$of fixed degree $n$ and bounded height $Q$ such that the derivative of theminimal polynomial $P_{\alpha}(x)$ of $\alpha$ is bounded,$|P_{\alpha}'(\alpha)| < Q^{1-v}$. This problem has many applications to theproblems of the metric theory of Diophantine approximation. We prove that thenumber of $\alpha$ defined above on the interval $\left(-\frac12,\frac12\right)$ doesn't exceed $c_1(n)Q^{n+1-\frac{1}{7}v}$ for $Q>Q_0(n)$ and$1.4 \le v \le \frac{7}{16}(n+1)$. Our result is based on an improvement to thelemma on the order of zero approximation by irreducible divisors of integerpolynomials from A. Gelfond's monograph "Transcendental and algebraic numbers".The improvement provides a stronger estimate for the absolute value of thedivisor in real points which are located far enough from all algebraic numbersof bounded degree and height and it's based on the representation of theresultant of two polynomials as the determinant of Sylvester matrix for theshifted polynomials. Keywords: Diophantine approximation, Hausdorff dimension,transcendental numbers, resultant, Sylvester matrix, irreducible divisor,Gelfond's lemma.

Counting real algebraic numbers with bounded derivative of minimal polynomial