derivative of (cos(ln(x))/x)

解

\frac{d}{d x} (\frac{cos ( ln ( x ) )}{x})

\frac{- sin ( ln ( x ) ) - cos ( ln ( x ) )}{x ^{2}}

解答ステップ

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

\frac{d}{d x} (cos (ln (x))) = - \frac{sin ( ln ( x ) )}{x}

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

= \frac{( - \frac{s i n ( l n ( x ) )}{x} ) x - 1 \cdot cos ( ln ( x ) )}{x ^{2}}

簡素化

\frac{( - \frac{s i n ( l n ( x ) )}{x} ) x - 1 \cdot cos ( ln ( x ) )}{x ^{2}} : \frac{- sin ( ln ( x ) ) - cos ( ln ( x ) )}{x ^{2}}

= \frac{- sin ( ln ( x ) ) - cos ( ln ( x ) )}{x ^{2}}