I understand how the sqrt comes down, from the right half of the top, but how does the top left part lose the square root?

After simplifying, I can get

sqrt(x^2+1) - x^2 / (x^2+1)^(3/2)

However, the answer that all online calcs give and my tutorial notes give is

1/ (x^2+1)^(3/2)

And I can see where that is going from the working up the top, but can't figure out how the sqrt(x^2+1) turns into (x^2+1)