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

解

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

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

解答ステップ

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

s

を定数として扱う

積の法則を適用:

(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

簡素化

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}