derivative of (x^2/(3x^2-2x+1))

解

\frac{d}{d x} (\frac{x ^{2}}{3 x ^{2} - 2 x + 1})

\frac{- 2 x ^{2} + 2 x}{( 3 x ^{2} - 2 x + 1 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{2}}{3 x ^{2} - 2 x + 1})

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} ) ( 3 x ^{2} - 2 x + 1 ) - \frac{d}{d x} ( 3 x ^{2} - 2 x + 1 ) x ^{2}}{( 3 x ^{2} - 2 x + 1 ) ^{2}}

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

\frac{d}{d x} (3 x^{2} - 2 x + 1) = 6 x - 2

= \frac{2 x ( 3 x ^{2} - 2 x + 1 ) - ( 6 x - 2 ) x ^{2}}{( 3 x ^{2} - 2 x + 1 ) ^{2}}

拡張

2 x (3 x^{2} - 2 x + 1) - (6 x - 2) x^{2} : - 2 x^{2} + 2 x

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