derivative of (ax^2+bx+ce^{(-x)/8})

解

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{8}})

\frac{- a e ^{- \frac{x}{8}} x ^{2} - b e ^{- \frac{x}{8}} x - c e ^{- \frac{x}{8}}}{8} + 2 a e^{- \frac{x}{8}} x + b e^{- \frac{x}{8}}

解答ステップ

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{8}})

簡素化

(a x^{2} + b x + c) e^{\frac{- x}{8}} : e^{- \frac{x}{8}} (a x^{2} + b x + c)

= \frac{d}{d x} (e^{- \frac{x}{8}} (a x^{2} + b x + c))

積の法則を適用:

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

f = e^{- \frac{x}{8}}, g = a x^{2} + b x + c

= \frac{d}{d x} (e^{- \frac{x}{8}}) (a x^{2} + b x + c) + \frac{d}{d x} (a x^{2} + b x + c) e^{- \frac{x}{8}}

\frac{d}{d x} (e^{- \frac{x}{8}}) = - \frac{1}{8} e^{- \frac{x}{8}}

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

= (- \frac{1}{8} e^{- \frac{x}{8}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{8}}

簡素化

(- \frac{1}{8} e^{- \frac{x}{8}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{8}} : \frac{- a e ^{- \frac{x}{8}} x ^{2} - b e ^{- \frac{x}{8}} x - c e ^{- \frac{x}{8}}}{8} + 2 a e^{- \frac{x}{8}} x + b e^{- \frac{x}{8}}

= \frac{- a e ^{- \frac{x}{8}} x ^{2} - b e ^{- \frac{x}{8}} x - c e ^{- \frac{x}{8}}}{8} + 2 a e^{- \frac{x}{8}} x + b e^{- \frac{x}{8}}