Differentiate the following functions with respect to x: x^2-x+1 … - Vidyadip

Question

Differentiate the following functions with respect to x:$\frac{\text{x}^2-\text{x}+1}{\text{x}^2+\text{x}+1}$

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Answer

We have,$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{x}^2-\text{x}+1}{\text{x}^2+\text{x}+1}\Big)$
Using quotient rule, we get
$\frac{(\text{x}^2+\text{x}+1)\frac{\text{d}}{\text{dx}}(\text{x}^2-\text{x}+1)-(\text{x}^2-\text{x}+1)\frac{\text{d}}{\text{dx}}(\text{x}^2+\text{x}+1)}{(\text{x}^2+\text{x}+1)^2}$
$=\frac{(\text{x}^2+\text{x}+1)(\text{2x}-1)-(\text{x}^2-\text{x}+1)(\text{2x}+1)}{(\text{x}^2+\text{x}+1)^2}$
$=\frac{(\text{x}^2+1-\text{x})(\text{2x}-1)-(\text{x}^2-\text{x}+1)(\text{2x}+1)}{(\text{x}^2+\text{x}+1)^2}$
$=\frac{\text{2x}^3+\text{2x}+\text{2x}^2-\text{x}^2-1-\text{x}-\text{2x}^3+\text{2x}^2-\text{2x}-\text{x}^2+\text{x}-1}{(\text{x}^2+\text{x}+1)^2}$
$=\frac{\text{2x}^2-2}{(\text{x}^2+\text{x}+1)^2}$
$=\frac{2(\text{x}^2-1)}{(\text{x}^2+\text{x}+1)^2}$

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