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

解

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

- \frac{9 x + 2}{2 ( x - 2 ) ^{3} 3 x - 1}

解答ステップ

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

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

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

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

簡素化

\frac{\frac{3}{2 3 x - 1} ( 2 - x ) ^{2} - ( - 2 ( 2 - x ) ) 3 x - 1}{( ( 2 - x ) ^{2} ) ^{2}} : - \frac{9 x + 2}{2 ( x - 2 ) ^{3} 3 x - 1}

= - \frac{9 x + 2}{2 ( x - 2 ) ^{3} 3 x - 1}