derivative (6x^2+1)/(x^2+9)

解

導関数

\frac{6 x ^{2} + 1}{x ^{2} + 9}

\frac{106 x}{( x ^{2} + 9 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{6 x ^{2} + 1}{x ^{2} + 9})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( 6 x ^{2} + 1 ) ( x ^{2} + 9 ) - \frac{d}{d x} ( x ^{2} + 9 ) ( 6 x ^{2} + 1 )}{( x ^{2} + 9 ) ^{2}}

\frac{d}{d x} (6 x^{2} + 1) = 12 x

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

= \frac{12 x ( x ^{2} + 9 ) - 2 x ( 6 x ^{2} + 1 )}{( x ^{2} + 9 ) ^{2}}

拡張

12 x (x^{2} + 9) - 2 x (6 x^{2} + 1) : 106 x

= \frac{106 x}{( x ^{2} + 9 ) ^{2}}