derivative f(x)=(x^2+8)/(9x-2)

解

導関数

f (x) = \frac{x ^{2} + 8}{9 x - 2}

\frac{9 x ^{2} - 4 x - 72}{( 9 x - 2 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2} + 8}{9 x - 2})

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} + 8 ) ( 9 x - 2 ) - \frac{d}{d x} ( 9 x - 2 ) ( x ^{2} + 8 )}{( 9 x - 2 ) ^{2}}

\frac{d}{d x} (x^{2} + 8) = 2 x

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

= \frac{2 x ( 9 x - 2 ) - 9 ( x ^{2} + 8 )}{( 9 x - 2 ) ^{2}}

拡張

2 x (9 x - 2) - 9 (x^{2} + 8) : 9 x^{2} - 4 x - 72

= \frac{9 x ^{2} - 4 x - 72}{( 9 x - 2 ) ^{2}}