derivative f(x)=((5x-2))/((x^2+1))

解

導関数

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

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

解答ステップ

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

商の法則を適用:

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

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

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

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

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

拡張

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

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