What is the derivative of this quotient

How would one differentiate x/x^2+1 ?
What is the formal procedure + simplification (please show working!!!)

Use quotient rule or product rule, whichever you prefer.
IF you use product rule you need to rewrite as (x)(x^2+1)^-1

Lets just use quotient rule:

[(x^2+1)(d/dxx) - (x)(d/dx x^2+1)] / (x^2 +1)^2

[(x^2+1)(1) - (x)(2x)] / (x^2+1)(x^2+1)

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

(-x^2 + 1) / (x^4+2x^2 + 1)

--- although its probably simpler to just leave the denominator as (x^2+1)^2