derivative of (ax^2+bx+c*e^{(-x)/4})

解答

\frac{d}{d x} ((a x^{2} + b x + c) \cdot e^{\frac{- x}{4}})

\frac{- a e ^{- \frac{x}{4}} x ^{2} - b e ^{- \frac{x}{4}} x - c e ^{- \frac{x}{4}}}{4} + 2 a e^{- \frac{x}{4}} x + b e^{- \frac{x}{4}}

求解步骤

\frac{d}{d x} ((a x^{2} + b x + c) e^{\frac{- x}{4}})

化简

(a x^{2} + b x + c) e^{\frac{- x}{4}} : e^{- \frac{x}{4}} (a x^{2} + b x + c)

= \frac{d}{d x} (e^{- \frac{x}{4}} (a x^{2} + b x + c))

使用微分乘法定则:

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

f = e^{- \frac{x}{4}}, g = a x^{2} + b x + c

= \frac{d}{d x} (e^{- \frac{x}{4}}) (a x^{2} + b x + c) + \frac{d}{d x} (a x^{2} + b x + c) e^{- \frac{x}{4}}

\frac{d}{d x} (e^{- \frac{x}{4}}) = - \frac{1}{4} e^{- \frac{x}{4}}

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

= (- \frac{1}{4} e^{- \frac{x}{4}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{4}}

化简

(- \frac{1}{4} e^{- \frac{x}{4}}) (a x^{2} + b x + c) + (2 a x + b) e^{- \frac{x}{4}} : \frac{- a e ^{- \frac{x}{4}} x ^{2} - b e ^{- \frac{x}{4}} x - c e ^{- \frac{x}{4}}}{4} + 2 a e^{- \frac{x}{4}} x + b e^{- \frac{x}{4}}

= \frac{- a e ^{- \frac{x}{4}} x ^{2} - b e ^{- \frac{x}{4}} x - c e ^{- \frac{x}{4}}}{4} + 2 a e^{- \frac{x}{4}} x + b e^{- \frac{x}{4}}

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