derivative f(x)=xe^{1/x}

解答

导数

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

e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}}}{x}

求解步骤

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

使用微分乘法定则:

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

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

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

\frac{d x}{d x} = 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

化简

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

= e^{\frac{1}{x}} - \frac{e ^{\frac{1}{x}}}{x}

a re a - cos (x), [0, π]

\frac{\partial}{\partial x} (2 y + x e^{y})

\frac{d y}{d t} = 0.06 y - 4000

t an g e n t 5 x^{2} + 2 x - 3

y^{^{'}} = \frac{x ^{2} \cdot e ^{(\frac{y}{x})} + y ^{2}}{x \cdot y}