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

Question

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

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Answer

$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{2x}-1}{\text{x}^2+1}\Big)=\frac{(\text{x}^2+1)\frac{\text{d}}{\text{dx}}(\text{2x}-1)-(\text{2x}-1)\frac{\text{d}}{\text{dx}}(\text{x}^2+1)}{(\text{x}^2+1)^2}$$=\frac{(\text{x}^2+1)\times2-(\text{2x}-1)\times\text{2x}}{(\text{x}^2+1)^2}$
$=\frac{\text{2x}^2+2-\text{4x}^2+\text{2x}}{(\text{x}^2+1)^2}$
$=\frac{-\text{2x}+\text{2x}+2}{(\text{x}^2+1)^2}$
$=\frac{2(-\text{x}^2+\text{x}+1)}{(\text{x}^2+1)^2}$
$=\frac{2(1+\text{x}-\text{x}^2)}{(\text{x}^2+1)^2}$

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