In the product rule you bring each of the two parts down twice, once differentiating it, and the other time not differentiating it. This makes for various possible ways of representing it. Here are two:
In the second of these, your problem gets each part differentiated on the ends of the bottom row, with the undifferentiated versions left in the middle - I think these were the bits you weren't sure of. (And remember the left-had one of the pair was tweaked - multiplied top and bottom by (x^2 - 1) for the sake of the common denominator.) Here's the simpler version that doesn't zoom in on the chain rule details...
Anyway, compare with the formula... Indeed, try this...