derivative of (sin(x^2)/(ln(2x+8)))

解

\frac{d}{d x} (\frac{sin ( x ^{2} )}{ln ( 2 x + 8 )})

\frac{2 x cos ( x ^{2} ) ln ( 2 x + 8 ) ( x + 4 ) - sin ( x ^{2} )}{ln ^{2} ( 2 x + 8 ) ( x + 4 )}

解答ステップ

\frac{d}{d x} (\frac{sin ( x ^{2} )}{ln ( 2 x + 8 )})

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 ^{2} ) ) ln ( 2 x + 8 ) - \frac{d}{d x} ( ln ( 2 x + 8 ) ) sin ( x ^{2} )}{( ln ( 2 x + 8 ) ) ^{2}}

\frac{d}{d x} (sin (x^{2})) = cos (x^{2}) \cdot 2 x

\frac{d}{d x} (ln (2 x + 8)) = \frac{1}{x + 4}

= \frac{cos ( x ^{2} ) \cdot 2 x ln ( 2 x + 8 ) - \frac{1}{x + 4} sin ( x ^{2} )}{( ln ( 2 x + 8 ) ) ^{2}}

簡素化

\frac{cos ( x ^{2} ) \cdot 2 x ln ( 2 x + 8 ) - \frac{1}{x + 4} sin ( x ^{2} )}{ln ^{2} ( 2 x + 8 )} : \frac{2 x cos ( x ^{2} ) ln ( 2 x + 8 ) ( x + 4 ) - sin ( x ^{2} )}{ln ^{2} ( 2 x + 8 ) ( x + 4 )}

= \frac{2 x cos ( x ^{2} ) ln ( 2 x + 8 ) ( x + 4 ) - sin ( x ^{2} )}{ln ^{2} ( 2 x + 8 ) ( x + 4 )}