derivative of (-ln(x+1)/(x^2))

解

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

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

解答ステップ

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

商の法則を適用:

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

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

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

\frac{d}{d x} (x^{2}) = 2 x

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

簡素化

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

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