(\partial)/(\partial t)(te^{s/t})

解答

\frac{\partial}{\partial t} (t e^{\frac{s}{t}})

e^{\frac{s}{t}} - \frac{e ^{\frac{s}{t}} s}{t}

求解步骤

\frac{\partial}{\partial t} (t e^{\frac{s}{t}})

将

s

视为常数

使用微分乘法定则:

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

f = t, g = e^{\frac{s}{t}}

= \frac{\partial}{\partial t} (t) e^{\frac{s}{t}} + \frac{\partial}{\partial t} (e^{\frac{s}{t}}) t

\frac{\partial}{\partial t} (t) = 1

\frac{\partial}{\partial t} (e^{\frac{s}{t}}) = - \frac{s e ^{\frac{s}{t}}}{t ^{2}}

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

化简

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

= e^{\frac{s}{t}} - \frac{e ^{\frac{s}{t}} s}{t}

\frac{d}{d x} (arctan (\frac{1 + x}{1 - x}))

d er i v a t i v e x^{3} - 3

d er i v a t i v e (1 + cos^{2} (x))^{5}

n = 0 \sum \infty \frac{( e ^{n} )}{5}

\frac{d}{d x} (- 8 x^{3})