Differentiate the following functions with respect to x :

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

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Solution

Using quotient rule, we have

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

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

$= \frac{(x^{2} + 1) \times 2 - (2 x - 1) \times 2 x}{(x^{2} + 1)^{2}} = \frac{2 x^{2} - 4 x^{2} + 2 x}{(x^{2} + 1)^{2}}$

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

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

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