Prove that the Legendre polynomial holds true using the generating function and a binomial expansion.

Question

I tried using that $$P_{2n}(z)=\frac1{2^{2n}(2n)!}\frac{d^{2n}}{dz^{2n}}(z^2-1)^{2n} $$ but I am having trouble taking the derivative when $n$ is unknown and $z$ is $0$. Any help would be greatly appreciated!