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

解

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

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

解答ステップ

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

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

\frac{d}{d x} (5 - 3 x) = - 3

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

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

簡素化

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

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