derivative of (x^2-8x+8e^{(9x)/8})

解

\frac{d}{d x} ((x^{2} - 8 x + 8) e^{\frac{9 x}{8}})

\frac{9 e ^{\frac{9 x}{8}} x ^{2}}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}

解答ステップ

\frac{d}{d x} ((x^{2} - 8 x + 8) e^{\frac{9 x}{8}})

積の法則を適用:

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

f = x^{2} - 8 x + 8, g = e^{\frac{9 x}{8}}

= \frac{d}{d x} (x^{2} - 8 x + 8) e^{\frac{9 x}{8}} + \frac{d}{d x} (e^{\frac{9 x}{8}}) (x^{2} - 8 x + 8)

\frac{d}{d x} (x^{2} - 8 x + 8) = 2 x - 8

\frac{d}{d x} (e^{\frac{9 x}{8}}) = e^{\frac{9 x}{8}} \frac{9}{8}

= (2 x - 8) e^{\frac{9 x}{8}} + e^{\frac{9 x}{8}} \frac{9}{8} (x^{2} - 8 x + 8)

簡素化

(2 x - 8) e^{\frac{9 x}{8}} + e^{\frac{9 x}{8}} \frac{9}{8} (x^{2} - 8 x + 8) : \frac{9 e ^{\frac{9 x}{8}} x ^{2}}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}

= \frac{9 e ^{\frac{9 x}{8}} x ^{2}}{8} - 7 e^{\frac{9 x}{8}} x + e^{\frac{9 x}{8}}