derivative of (1+sqrt(x)/x)

解

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

\frac{- x - 2}{2 x ^{2}}

解答ステップ

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

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

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

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

簡素化

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

= \frac{- x - 2}{2 x ^{2}}