What is the derivative of te^{(-t)/2} ?

The derivative of te^{(-t)/2} is -(e^{-t/2}t)/2+e^{-t/2}

What is the first derivative of te^{(-t)/2} ?

The first derivative of te^{(-t)/2} is -(e^{-t/2}t)/2+e^{-t/2}

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derivative of

t e^{\frac{- t}{2}}

- \frac{e ^{- \frac{t}{2}} t}{2} + e^{- \frac{t}{2}}

Solution steps

\frac{d}{d t} (t e^{\frac{- t}{2}})

Simplify

t e^{\frac{- t}{2}} : e^{- \frac{t}{2}} t

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

Apply the Product Rule:

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

f = e^{- \frac{t}{2}}, g = t

= \frac{d}{d t} (e^{- \frac{t}{2}}) t + \frac{d t}{d t} e^{- \frac{t}{2}}

\frac{d}{d t} (e^{- \frac{t}{2}}) = - \frac{1}{2} e^{- \frac{t}{2}}

\frac{d t}{d t} = 1

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

Simplify

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

= - \frac{e ^{- \frac{t}{2}} t}{2} + e^{- \frac{t}{2}}

What is the derivative of te^{(-t)/2} ?
The derivative of te^{(-t)/2} is -(e^{-t/2}t)/2+e^{-t/2}
What is the first derivative of te^{(-t)/2} ?
The first derivative of te^{(-t)/2} is -(e^{-t/2}t)/2+e^{-t/2}