derivative of x/(x+b/x)

解

\frac{d}{d x} (\frac{x}{x + \frac{b}{x}})

\frac{2 b x}{( x ^{2} + b ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x}{x + \frac{b}{x}})

商の法則を適用:

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

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

\frac{d x}{d x} = 1

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

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

簡素化

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

= \frac{2 b x}{( x ^{2} + b ) ^{2}}