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Aus Online Mathematik Brückenkurs 2

The expression is a quotient of two polynomials, x2+1 and x+1, and we therefore use the quotient rule for differentiation,

x+1x2+1=(x+1)2(x2+1)(x+1)−(x2+1)(x+1)=(x+1)22x(x+1)−(x2+1)1=(x+1)22x2+2x−x2−1=(x+1)2x2+2x−1.

Note: It is possible to rewrite the numerator by completing the square,

x2+2x−1=(x+1)2−12−1=(x+1)2−2

and then the answer can be written as

(x+1)2x2+2x−1=(x+1)2(x+1)2−2=1−2(x+1)2.