derivative y=((2x-3)/((x^2+4)^2))

解

導関数

y = (\frac{2 x - 3}{( x ^{2} + 4 ) ^{2}})

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

解答ステップ

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

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

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

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

簡素化

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

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