derivative of (x-5/(x^2-3x+15))

解

\frac{d}{d x} (\frac{x - 5}{x ^{2} - 3 x + 15})

\frac{- x ^{2} + 10 x}{( x ^{2} - 3 x + 15 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x - 5}{x ^{2} - 3 x + 15})

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 - 5 ) ( x ^{2} - 3 x + 15 ) - \frac{d}{d x} ( x ^{2} - 3 x + 15 ) ( x - 5 )}{( x ^{2} - 3 x + 15 ) ^{2}}

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

\frac{d}{d x} (x^{2} - 3 x + 15) = 2 x - 3

= \frac{1 \cdot ( x ^{2} - 3 x + 15 ) - ( 2 x - 3 ) ( x - 5 )}{( x ^{2} - 3 x + 15 ) ^{2}}

簡素化

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

= \frac{- x ^{2} + 10 x}{( x ^{2} - 3 x + 15 ) ^{2}}