Bernstein inequality for algebraic polynomials ($|p^{\prime }(x)|\le n\Vert p{\Vert }_{[-1,1]}/\sqrt{1-x^2}$)

For a polynomial of degree at most $n$, this derivative estimate holds for $-1<x<1$. It controls the derivative away from the endpoints. Sampling it at Chebyshev polynomial roots supplies the hypotheses of the Chebyshev nodal derivative comparison, which controls the missing endpoint regions. The norm is taken on the entire interval, although the displayed pointwise bound concerns its interior.