derivative of-(e^{x-1}(x-3)/((x-1)^3))

解

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

- \frac{e ^{x - 1} ( x ^{2} - 6 x + 11 )}{( x - 1 ) ^{4}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

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

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

キャンセル

\frac{( e ^{x - 1} x - 2 e ^{x - 1} ) ( x - 1 ) ^{3} - 3 ( x - 1 ) ^{2} e ^{x - 1} ( x - 3 )}{( ( x - 1 ) ^{3} ) ^{2}} : \frac{e ^{x - 1} ( x ^{2} - 6 x + 11 )}{( x - 1 ) ^{4}}

= - \frac{e ^{x - 1} ( x ^{2} - 6 x + 11 )}{( x - 1 ) ^{4}}