derivative of-(e^{-x}(x^3+3(x^2+2x+2))/(x^4))

解

\frac{d}{d x} (- \frac{e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{x ^{4}})

\frac{e ^{- x} ( x ^{4} + 4 ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) )}{x ^{5}}

解答ステップ

\frac{d}{d x} (- \frac{e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{x ^{4}})

定数を除去:

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

= - \frac{d}{d x} (\frac{e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{x ^{4}})

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} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) ) x ^{4} - \frac{d}{d x} ( x ^{4} ) e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{( x ^{4} ) ^{2}}

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

\frac{d}{d x} (x^{4}) = 4 x^{3}

= - \frac{( - e ^{- x} x ^{3} ) x ^{4} - 4 x ^{3} e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{( x ^{4} ) ^{2}}

簡素化

- \frac{( - e ^{- x} x ^{3} ) x ^{4} - 4 x ^{3} e ^{- x} ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) )}{( x ^{4} ) ^{2}} : \frac{e ^{- x} ( x ^{4} + 4 ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) )}{x ^{5}}

= \frac{e ^{- x} ( x ^{4} + 4 ( x ^{3} + 3 ( x ^{2} + 2 x + 2 ) ) )}{x ^{5}}