derivative of ((x^2+1^3)/((4x-5)^5))

解

\frac{d}{d x} (\frac{( x ^{2} + 1 ) ^{3}}{( 4 x - 5 ) ^{5}})

\frac{2 ( x ^{2} + 1 ) ^{2} ( 2 x ^{2} - 15 x - 10 )}{( 4 x - 5 ) ^{6}}

解答ステップ

\frac{d}{d x} (\frac{( x ^{2} + 1 ) ^{3}}{( 4 x - 5 ) ^{5}})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( ( x ^{2} + 1 ) ^{3} ) ( 4 x - 5 ) ^{5} - \frac{d}{d x} ( ( 4 x - 5 ) ^{5} ) ( x ^{2} + 1 ) ^{3}}{( ( 4 x - 5 ) ^{5} ) ^{2}}

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

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

= \frac{6 x ( x ^{2} + 1 ) ^{2} ( 4 x - 5 ) ^{5} - 20 ( 4 x - 5 ) ^{4} ( x ^{2} + 1 ) ^{3}}{( ( 4 x - 5 ) ^{5} ) ^{2}}

簡素化

\frac{6 x ( x ^{2} + 1 ) ^{2} ( 4 x - 5 ) ^{5} - 20 ( 4 x - 5 ) ^{4} ( x ^{2} + 1 ) ^{3}}{( ( 4 x - 5 ) ^{5} ) ^{2}} : \frac{2 ( x ^{2} + 1 ) ^{2} ( 2 x ^{2} - 15 x - 10 )}{( 4 x - 5 ) ^{6}}

= \frac{2 ( x ^{2} + 1 ) ^{2} ( 2 x ^{2} - 15 x - 10 )}{( 4 x - 5 ) ^{6}}