derivative of (ln(3x^2-6x)/(4x^3-2))

解

\frac{d}{d x} (\frac{ln ( 3 x ^{2} - 6 x )}{4 x ^{3} - 2})

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

解答ステップ

\frac{d}{d x} (\frac{ln ( 3 x ^{2} - 6 x )}{4 x ^{3} - 2})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( ln ( 3 x ^{2} - 6 x ) ) ( 4 x ^{3} - 2 ) - \frac{d}{d x} ( 4 x ^{3} - 2 ) ln ( 3 x ^{2} - 6 x )}{( 4 x ^{3} - 2 ) ^{2}}

\frac{d}{d x} (ln (3 x^{2} - 6 x)) = \frac{2 ( x - 1 )}{x ( x - 2 )}

\frac{d}{d x} (4 x^{3} - 2) = 12 x^{2}

= \frac{\frac{2 ( x - 1 )}{x ( x - 2 )} ( 4 x ^{3} - 2 ) - 12 x ^{2} ln ( 3 x ^{2} - 6 x )}{( 4 x ^{3} - 2 ) ^{2}}

簡素化

\frac{\frac{2 ( x - 1 )}{x ( x - 2 )} ( 4 x ^{3} - 2 ) - 12 x ^{2} ln ( 3 x ^{2} - 6 x )}{( 4 x ^{3} - 2 ) ^{2}} : \frac{( x - 1 ) ( 3 2 x - 1 ) ( 2 ^{\frac{2}{3}} x ^{2} + 3 2 x + 1 ) - 3 x ^{3} ln ( 3 x ( x - 2 ) ) ( x - 2 )}{x ( x - 2 ) ( 3 2 x - 1 ) ^{2} ( 2 ^{\frac{2}{3}} x ^{2} + 3 2 x + 1 ) ^{2}}

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