derivative of (ax^2+bx+ce^{-x/6})

解

\frac{d}{d x} ((a x^{2} + b x + c) e^{- \frac{x}{6}})

2 a e^{- \frac{x}{6}} x + b e^{- \frac{x}{6}} + \frac{- a e ^{- \frac{x}{6}} x ^{2} - b e ^{- \frac{x}{6}} x - c e ^{- \frac{x}{6}}}{6}

解答ステップ

\frac{d}{d x} ((a x^{2} + b x + c) e^{- \frac{x}{6}})

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = a x^{2} + b x + c, g = e^{- \frac{x}{6}}

= \frac{d}{d x} (a x^{2} + b x + c) e^{- \frac{x}{6}} + \frac{d}{d x} (e^{- \frac{x}{6}}) (a x^{2} + b x + c)

\frac{d}{d x} (a x^{2} + b x + c) = 2 a x + b

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

= (2 a x + b) e^{- \frac{x}{6}} + (- \frac{1}{6} e^{- \frac{x}{6}}) (a x^{2} + b x + c)

簡素化

(2 a x + b) e^{- \frac{x}{6}} + (- \frac{1}{6} e^{- \frac{x}{6}}) (a x^{2} + b x + c) : 2 a e^{- \frac{x}{6}} x + b e^{- \frac{x}{6}} + \frac{- a e ^{- \frac{x}{6}} x ^{2} - b e ^{- \frac{x}{6}} x - c e ^{- \frac{x}{6}}}{6}

= 2 a e^{- \frac{x}{6}} x + b e^{- \frac{x}{6}} + \frac{- a e ^{- \frac{x}{6}} x ^{2} - b e ^{- \frac{x}{6}} x - c e ^{- \frac{x}{6}}}{6}