derivative x/(x^2+13x+40)

解

導関数

\frac{x}{x ^{2} + 13 x + 40}

\frac{- x ^{2} + 40}{( x ^{2} + 13 x + 40 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x}{x ^{2} + 13 x + 40})

商の法則を適用:

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

= \frac{\frac{d x}{d x} ( x ^{2} + 13 x + 40 ) - \frac{d}{d x} ( x ^{2} + 13 x + 40 ) x}{( x ^{2} + 13 x + 40 ) ^{2}}

\frac{d x}{d x} = 1

\frac{d}{d x} (x^{2} + 13 x + 40) = 2 x + 13

= \frac{1 \cdot ( x ^{2} + 13 x + 40 ) - ( 2 x + 13 ) x}{( x ^{2} + 13 x + 40 ) ^{2}}

簡素化

\frac{1 \cdot ( x ^{2} + 13 x + 40 ) - ( 2 x + 13 ) x}{( x ^{2} + 13 x + 40 ) ^{2}} : \frac{- x ^{2} + 40}{( x ^{2} + 13 x + 40 ) ^{2}}

= \frac{- x ^{2} + 40}{( x ^{2} + 13 x + 40 ) ^{2}}