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

解

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

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

解答ステップ

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

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

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

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

簡素化

\frac{( - sin ( x ) ) ln ( x ) - \frac{1}{x} cos ( x )}{ln ^{2} ( x )} : \frac{- x sin ( x ) ln ( x ) - cos ( x )}{x ln ^{2} ( x )}

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