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Differentiation preserves the ordering of interlacing roots of polynomials of equal or consecutive polynomial degrees. Iteration gives the result for every derivative for which both polynomials are nonconstant; common roots are allowed by a limiting argument. This is a result about the locations of roots of a polynomial, distinct from the Markov inequality for polynomial derivatives. Theorem A of Dimitar K. Dimitrov's account states the first-derivative result.

The Markov interlacing lemma says that differentiation preserves interlacing roots of polynomials. More explicitly, if two degree-$d$ real polynomials have ordered real roots satisfying $a_1\le b_1\le a_2\le \cdots \le a_d\le b_d$, their $k$th derivatives have ordered roots $a_j^{(k)},b_j^{(k)}$ satisfyingThe analogous alternating order is preserved for polynomials of consecutive degrees. Common roots are allowed through weak inequalities; strictly interlacing simple roots give strict inequalities. When a derivative is constant, its root list is empty.