First Derivative of Legendre Polynomials

Question

I have been tasked with calculating the first derivative of the k-th Legendre polynomial $$P^{'}_k(1)$$ I was given the hint to use the generalized product rule $$\frac{d^{n}}{ds^{n}}[F(s)G(s)]=\sum_{j=0}^n {n \choose j}F^{(n-j)}(s)G^{(j)}(s)$$ but am not exactly sure how to use this. I started by using Rodrigues' formula $$P_k(s)=\frac{1}{2^{k}k!}(\frac{d}{ds})^{k}(s^2-1)^k$$ I then noticed that $$(s^2-1)=(s-1)(s+1)$$ which means that $$P_k(s)=\frac{1}{2^{k}k!}(\frac{d}{ds})^{k}(s-1)^k(s+1)^k$$ I'm not sure where to go from here or if this is even a beneficial route to take. Any advice/hints to get to the final calculation would be much appreciated!

Answer 1

Hint: Take $n=k+1,$ $G(s)=(s0-1)^k$ and $F(s)=(s+1)^k$So, you don't want to leave any $(s-1)^j$ with $j>0,$ hence the only term that will not vanish at $1$ is when $j=k.$ Check that $\frac{d^k}{ds^k}(s-1)^k=k!$ and use your formula, and the fact that $\frac{d}{ds}(s+1)^k=k2^{k-1}.$

You should get $\frac{k(k+1)}{2}.$