derivative (x^2-1)/((2x-5)^2)

解

導関数

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

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

解答ステップ

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

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

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

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

簡素化

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

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