derivative f(x)=(x+2)e^{1/x}

解

導関数

f (x) = (x + 2) e^{\frac{1}{x}}

e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}} ( x + 2 )}{x ^{2}}

解答ステップ

\frac{d}{d x} ((x + 2) e^{\frac{1}{x}})

積の法則を適用:

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

f = x + 2, g = e^{\frac{1}{x}}

= \frac{d}{d x} (x + 2) e^{\frac{1}{x}} + \frac{d}{d x} (e^{\frac{1}{x}}) (x + 2)

\frac{d}{d x} (x + 2) = 1

\frac{d}{d x} (e^{\frac{1}{x}}) = - \frac{e ^{\frac{1}{x}}}{x ^{2}}

= 1 \cdot e^{\frac{1}{x}} + (- \frac{e ^{\frac{1}{x}}}{x ^{2}}) (x + 2)

簡素化

1 \cdot e^{\frac{1}{x}} + (- \frac{e ^{\frac{1}{x}}}{x ^{2}}) (x + 2) : e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}} ( x + 2 )}{x ^{2}}

= e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}} ( x + 2 )}{x ^{2}}