Answer to Question #145324 in Calculus for Abdulsalam Aminu Bandam

Question

Find dy/dx and d^2y/dx^2 if y=1+x/1+x^2

Answer 1

$y = \cfrac{1+x}{1+x^2}$

$y^\prime = \cfrac{\partial y}{\partial x} = \cfrac{1+x^2 - 2x(1+x)}{(1+x^2)^2} =\\= \cfrac{1-2x-x^2}{(1+x^2)^2}$

$y^{\prime\prime} = \cfrac{\partial^2y}{\partial x^2} = \\ = \cfrac{(-2-2x)(1+x^2)^2 - 2(1+x^2)2x(1-2x-x^2)}{(1+x^2)^4}= \\= \cfrac{-2(1+x)(1+x^2)-4x(1-2x-x^2)}{(1+x^2)^3}=\\= \cfrac{-2((1+x)(1+x^2) +2x(1-2x-x^2))}{(1+x^2)^3} =\\= \cfrac{-2(1+x^2+x+x^3+2x-4x^2-2x^3)}{(1+x^2)^3}=\\= \cfrac{-2-6x+6x^2+2x^3}{(1+x^2)^3}$