I have been trying to prove the relation$\left (P'_n(1)=\frac{1}{2}n(n+1)\right )$ and have proved it directly from the Legendre differential equation as $P_n(x)${the Legendre polynomial} is the solution of the Legendre differential equation, given as $$(1-x^2)\frac{d^2y}{dx^2}-2x\frac{dy}{dx}+n(n+1)y=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...(1)$$and we can write equation $(1)$ as $$(1-x^2)P''_n(x)-2xP'_n(x)+n(n+1)P_n(x)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...(2)$$and then putting $x=1$ in $(2)$, we get $$0\:-\:2P'_n(1)\:+\: n(n+1)P_n(1)=0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:...(3)$$ $$\implies 2P'_n(1)=n(n+1)\:.(1)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:[as P_n(1)=1]$$ $$\implies P'_n(1)=\frac{1}{2}n(n+1)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$where $P'_n(x)$ and $P''_n(x)$ are first order and second order derivative of Legendre polynomial respectively.
But my question is how can we prove the same relation by using the Rodrigues's Formula which is given as follows $$P_n(x)=\frac{1}{2^n.n!}\frac{d^n}{dx^n}(x^2-1)^n$$ I have been trying to solve the relation by using Rodrigues's Formula but I don't have any idea how to solve it. Please help.
Thanks beforehand :)