derivative of ((sin(1/x*x^3))/(ln(x)))

解

\frac{d}{d x} (\frac{( sin ( \frac{1}{x} ) \cdot x ^{3} )}{ln ( x )})

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

解答ステップ

\frac{d}{d x} (\frac{( sin ( \frac{1}{x} ) x ^{3} )}{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 ( \frac{1}{x} ) x ^{3} ) ln ( x ) - \frac{d}{d x} ( ln ( x ) ) sin ( \frac{1}{x} ) x ^{3}}{( ln ( x ) ) ^{2}}

\frac{d}{d x} (sin (\frac{1}{x}) x^{3}) = - x cos (\frac{1}{x}) + 3 x^{2} sin (\frac{1}{x})

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

= \frac{( - x cos ( \frac{1}{x} ) + 3 x ^{2} sin ( \frac{1}{x} ) ) ln ( x ) - \frac{1}{x} sin ( \frac{1}{x} ) x ^{3}}{( ln ( x ) ) ^{2}}

\frac{1}{x} sin (\frac{1}{x}) x^{3} = x^{2} sin (\frac{1}{x})

= \frac{ln ( x ) ( - x cos ( \frac{1}{x} ) + 3 x ^{2} sin ( \frac{1}{x} ) ) - x ^{2} sin ( \frac{1}{x} )}{ln ^{2} ( x )}