derivative y=(x+1/x)/(2sqrt(x)+1/x)

解

導関数

y = \frac{x + \frac{1}{x}}{2 x + \frac{1}{x}}

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

解答ステップ

\frac{d}{d x} (\frac{x + \frac{1}{x}}{2 x + \frac{1}{x}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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

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