Question

derivate x2+1/x+1 by using formula d/dx (u/v)=v du/dx - u dv/dx /v2

Answer 1

u = (x² + 1)            du /dx  = 2x
v = x + 1              dv/dx = 1

$u = x^2 + 1,\ \ \ \ \ \frac{du}{dx}=2x\\ \\v = x + 1,\ \ \ \ \frac{dv}{dx}=1\\ \\\frac{d\ (u/v)}{dx}=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}=\frac{(x+1)2x-(x^2+1)1}{(x+1)^2}\\ \\=\frac{2x^2+2x-x^2-1}{(x+1)^2}=\frac{(x+1)^2-2}{(x+1)^2}=1 - \frac{2}{(x+1)^2}=\frac{x^2+2x-1}{(x+1)^2}$