derivative of ((4x-4^5)/((2x-5)^7))

解

\frac{d}{d x} (\frac{( 4 x - 4 ) ^{5}}{( 2 x - 5 ) ^{7}})

\frac{4 ( 4 x - 4 ) ^{4} ( - 4 x - 11 )}{( 2 x - 5 ) ^{8}}

解答ステップ

\frac{d}{d x} (\frac{( 4 x - 4 ) ^{5}}{( 2 x - 5 ) ^{7}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( ( 4 x - 4 ) ^{5} ) ( 2 x - 5 ) ^{7} - \frac{d}{d x} ( ( 2 x - 5 ) ^{7} ) ( 4 x - 4 ) ^{5}}{( ( 2 x - 5 ) ^{7} ) ^{2}}

\frac{d}{d x} ((4 x - 4)^{5}) = 20 (4 x - 4)^{4}

\frac{d}{d x} ((2 x - 5)^{7}) = 14 (2 x - 5)^{6}

= \frac{20 ( 4 x - 4 ) ^{4} ( 2 x - 5 ) ^{7} - 14 ( 2 x - 5 ) ^{6} ( 4 x - 4 ) ^{5}}{( ( 2 x - 5 ) ^{7} ) ^{2}}

簡素化

\frac{20 ( 4 x - 4 ) ^{4} ( 2 x - 5 ) ^{7} - 14 ( 2 x - 5 ) ^{6} ( 4 x - 4 ) ^{5}}{( ( 2 x - 5 ) ^{7} ) ^{2}} : \frac{4 ( 4 x - 4 ) ^{4} ( - 4 x - 11 )}{( 2 x - 5 ) ^{8}}

= \frac{4 ( 4 x - 4 ) ^{4} ( - 4 x - 11 )}{( 2 x - 5 ) ^{8}}