In a homework question, we are asked to show that the Legendre polynomials do indeed solve the Legendre Differential Equation: $$ \frac{d}{dx} \left[ (1 - x^2) \frac{d}{dx} P_n(x) \right] + n (n + 1) P_n(x) = 0. $$ According to Wikipedia, it is sufficient to show that after deriving it n+1 times the following equation must hold true: $$ (x^2 - 1) \frac{d}{dx} (x^2 - 1)^n = 2 n x (x^2 - 1)^n. $$ This was also given as a hint in the problem. Since I don't want to blindly accept the hint, I was trying to figure out how it was deduced, but didn't manage to. My question therefore is, how do we arrive at the equation above (the "hint equation")?
I can see, that the left part of the equation is nearly equal to the first part of the Legendre Differential Equation, except for a missing outer derivative. However, I am somehow missing the steps taken to arrive at the right side: Where does the $2nx$ come from?