Let #160;u=2x 1; #160;v=x2+1;Then, #160;u #39;=2; #160;v #39;=2xUsing #160;the #160;quotient #160;rule:ddxuv=vu #39; uv #39;v2ddx2x 1x2+1= ...

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Inequalities Involving Mathematical Means

2 x-1 x 2+1

Question

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

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Solution

$Let u = 2 x - 1; v = x^{2} + 1; Then, u' = 2; v' = 2 x Using the quotient rule: \frac{d}{d x} (\frac{u}{v}) = \frac{v u' - u v'}{v^{2}} \frac{d}{d x} (\frac{2 x - 1}{x^{2} + 1}) = \frac{(x^{2} + 1) 2 - (2 x - 1) 2 x}{(x^{2} + 1)^{2}} = \frac{2 x^{2} + 2 - 4 x^{2} + 2 x}{(x^{2} + 1)^{2}} = \frac{- 2 x^{2} + 2 x + 2}{(x^{2} + 1)^{2}} = \frac{2 (1 + x - x^{2})}{(x^{2} + 1)^{2}}$

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