I expect you are meant to use the quotient rule to do this, but since

I don't lnow it I will use the product rule:

df/dx= 1/(x^2+1) + x (-1)(2x)/(x^2+1)^2 = (x^2+1-2x^2)/(x^2+1)^2

.............= (1-x^2)/(x^2+1)^2

so:

d^2f/dx^2 = -2x/(x^2+1)^2 + (1-x^2) (-2) (2x)/(x^2+1)^3

..............= [-2x(x^2+1)=4x+4x^3]/(x^2+1)^3

..............= [2x(x^2=3)]/(x^2+1)^3

RonL