derivative of (sin(x)/(ln(x)))

解

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

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

解答ステップ

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

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

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

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

簡素化

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

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