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

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

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

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

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

Solution steps

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

Treat

w

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

\int x (4 x + 1) d x

\frac{df}{d t} = 4 t + \frac{9 t}{9 - t ^{2}}

x \cdot y^{2}

x \to 3 lim (- x + 3)

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

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