for the final answer it is 2x instead of x^2:
dy/dy = 2x(1/2)(x^2+1)^(-1/2)
take for example X^n
differentiating it w.r.t x will give:
n(d/dx X)(X)^(n-1)
take for example (X+1)^n
differentiating it w.r.t x will give:
n(d/dx (X+1))(X+1)^(n-1)
take for example ((X^m)+1)^n
differentiating it w.r.t x will give:
n(d/dx ((X^m)+1))((X^m)+1)^(n-1)
in youe question n = 1/2, m=2.