derivative y=((2x^2-1)^3)/((5x-8)^2)

解

導関数

y = \frac{( 2 x ^{2} - 1 ) ^{3}}{( 5 x - 8 ) ^{2}}

\frac{2 ( 2 x ^{2} - 1 ) ^{2} ( 20 x ^{2} - 48 x + 5 )}{( 5 x - 8 ) ^{3}}

解答ステップ

\frac{d}{d x} (\frac{( 2 x ^{2} - 1 ) ^{3}}{( 5 x - 8 ) ^{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} ( ( 2 x ^{2} - 1 ) ^{3} ) ( 5 x - 8 ) ^{2} - \frac{d}{d x} ( ( 5 x - 8 ) ^{2} ) ( 2 x ^{2} - 1 ) ^{3}}{( ( 5 x - 8 ) ^{2} ) ^{2}}

\frac{d}{d x} ((2 x^{2} - 1)^{3}) = 12 x (2 x^{2} - 1)^{2}

\frac{d}{d x} ((5 x - 8)^{2}) = 10 (5 x - 8)

= \frac{12 x ( 2 x ^{2} - 1 ) ^{2} ( 5 x - 8 ) ^{2} - 10 ( 5 x - 8 ) ( 2 x ^{2} - 1 ) ^{3}}{( ( 5 x - 8 ) ^{2} ) ^{2}}

簡素化

\frac{12 x ( 2 x ^{2} - 1 ) ^{2} ( 5 x - 8 ) ^{2} - 10 ( 5 x - 8 ) ( 2 x ^{2} - 1 ) ^{3}}{( ( 5 x - 8 ) ^{2} ) ^{2}} : \frac{2 ( 2 x ^{2} - 1 ) ^{2} ( 20 x ^{2} - 48 x + 5 )}{( 5 x - 8 ) ^{3}}

= \frac{2 ( 2 x ^{2} - 1 ) ^{2} ( 20 x ^{2} - 48 x + 5 )}{( 5 x - 8 ) ^{3}}