derivative f(x)=(x^2-9)/(x^2+3x)

解

導関数

f (x) = \frac{x ^{2} - 9}{x ^{2} + 3 x}

\frac{3}{x ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} - 9}{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} ( x ^{2} - 9 ) ( x ^{2} + 3 x ) - \frac{d}{d x} ( x ^{2} + 3 x ) ( x ^{2} - 9 )}{( x ^{2} + 3 x ) ^{2}}

\frac{d}{d x} (x^{2} - 9) = 2 x

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

= \frac{2 x ( x ^{2} + 3 x ) - ( 2 x + 3 ) ( x ^{2} - 9 )}{( x ^{2} + 3 x ) ^{2}}

簡素化

\frac{2 x ( x ^{2} + 3 x ) - ( 2 x + 3 ) ( x ^{2} - 9 )}{( x ^{2} + 3 x ) ^{2}} : \frac{3}{x ^{2}}

= \frac{3}{x ^{2}}