What is (\partial)/(\partial t)(te^{s/t}) ?

The answer to (\partial)/(\partial t)(te^{s/t}) is e^{s/t}-(e^{s/t}s)/t

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

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

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

Solution steps

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

Treat

s

as a constant

Apply the Product Rule:

(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

Simplify

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}

What is (\partial)/(\partial t)(te^{s/t}) ?
The answer to (\partial)/(\partial t)(te^{s/t}) is e^{s/t}-(e^{s/t}s)/t