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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.

Apply the Bernstein inequality for algebraic polynomials to the central interval $|x|\le \cos (\pi /(2n))$, using $\sin (\pi /(2n))\ge 1/n$. On the remaining regions apply the Chebyshev nodal derivative comparison to $p^{\prime }$. Since $T_n^{\prime }(\cos \theta )=n\sin (n\theta )/\sin \theta$, the identity expressing $\sin (n\theta )/\sin \theta$ as a sum of $n$ unit complex phases gives $|T_n^{\prime }|\le n^2$. The endpoint limit is $T_n^{\prime }(1)=n^2$, so the constant is sharp.