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

解

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

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

解答ステップ

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

簡素化

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

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

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

商の法則を適用:

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

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

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

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

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

簡素化

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

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