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

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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