Simple Quotient Rule?

Question

What is the derivative of 1+2x/(1+x^2)
The answer does not accord with my normally flawless mental computations...plus I don't see how it could have been arrived at...its 2-c/(1+x^2)^(3/2)...
Huh..?
First principles, anyone..?

Answer 1

Reading as:-
f (x) = (1 + 2x) / (1 + x²)
f `(x) = [ (1 + x²)(2) - (1 + 2x)(2x) ] / (1 + x²)²
f `(x) = [2 + 2x² - 2x - 4x²] / (1 + x²)²
f `(x) = [ 2 - 2x - 2x² ] / (1 + x²)²
f `(x) = [ 2 (1 - x - x² ] / (1 + x²)²

However you may mean:-
f (x) = 1 + (2x)(1 + x²)^(-1)
f `(x) = 2 (1 + x²)^(-1) + (-1)(1 + x²)^(-2)(2x)(2x)
f `(x) = 2 / (1 + x²) - 4x² / (1 + x²)²
f `(x) = [2 + 2x² - 4x²] / (1 + x²)²
f `(x) = (2 - 2x²) / (1 + x²)²
f `(x) = 2 (1 - x²) / (1 + x²)²

Would not use 1st principles for this.

Answer 2

Quotient rule is d(F\H) = (HF' -FH') / H^2. So I got, (-2x^2 - 2x
+ 2) / (x^4 + 2x^2 + 1)