= Derivative orthogonality when the Sturm-Liouville potential vanishes
{title2=$\int_a^bp\,y_n'y_m'\,dx=\lambda_n\int_a^bw\,y_ny_m\,dx$}
For the <Sturm-Liouville problem> $-(py_n')'=\lambda_nwy_n$, assume that the boundary term $[py_n'y_m]_a^b$ vanishes. One <integration by parts> gives the displayed identity. Distinct <eigenvalues> make the right side zero by <orthogonality of Sturm-Liouville eigenfunctions>, so the derivatives are orthogonal with weight $p$, while the original functions are orthogonal with weight $w$. This conclusion uses the absence of the $qy$ term: for a general potential the extra integral $-\int qy_ny_m$ need not vanish. For the <Chebyshev polynomial of the second kind> of degree $n-1$, the two weights are $w=\sqrt{1-x^2}$ and $p=(1-x^2)^{3/2}$.
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