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Differentiate the Legendre differential equation once and put $R_n=P_n^{\prime }$:Multiplying by $1-x^2$ gives its Sturm-Liouville form,Thus the weight is $w(x)=1-x^2$, not one. For $n\ge 1$, $R_n$ is a nonzero polynomial; $R_0=0$ is a trivial solution rather than a nonzero eigenfunction.

Multiply the equations for $n,m$ by $R_m,R_n$ respectively, subtract and integrate. The resulting boundary term is $[(1-x^2{)}^2(R_n^{\prime }{R}_m-R_m^{\prime }{R}_n){]}_{-1}^1=0$, since all these derivatives are finite at the endpoints. For distinct nonnegative indices, ${\lambda }_n-{\lambda }_m\ne 0$, giving the weighted orthogonality of Legendre polynomial derivatives:For $n=0$ the integral is zero simply because $R_0=0$.