Question

Differentiate the following functions with respect to x :

$\frac{x^2-x+1}{x^2+x+1}$

Answer 1

We have,

$\frac{d}{dx}\left(\frac{x^2-x+1}{x^2+x+1}\right)$

Using quotient rule,

$\frac{(x^2+x+1)\frac{d}{dx}(x^2-x+1)-(x^2-x-1)\frac{d}{dx}(x^2+x+1)}{(x^2+x+1{)}^2}$

$=\frac{(x^2+x+1)(2x-1)-(x^2-x+1)(2x+1)}{(x^2+x+1)}$

$=\frac{(x^2+1-x)(2x-1)-(x^2-x+1)(2x+1)}{(x^2+x+1{)}^2}$

$=\frac{2x^3+2x+2x^2-x^2-1-x-2x^3+2x^2-2x-x^2+x-1}{(x^2+x+1{)}^2}$

$=\frac{2x^2-2}{(x^2+x+1{)}^2}=\frac{2(x^2-1)}{(x^2+x+1{)}^2}$