What is the derivative of xe^{-x/2} ?

The derivative of xe^{-x/2} is e^{-x/2}-(e^{-x/2}x)/2

What is the first derivative of xe^{-x/2} ?

The first derivative of xe^{-x/2} is e^{-x/2}-(e^{-x/2}x)/2

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{d x}{d x} = 1

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

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

Simplify

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

= e^{- \frac{x}{2}} - \frac{e ^{- \frac{x}{2}} x}{2}

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What is the derivative of xe^{-x/2} ?
The derivative of xe^{-x/2} is e^{-x/2}-(e^{-x/2}x)/2
What is the first derivative of xe^{-x/2} ?
The first derivative of xe^{-x/2} is e^{-x/2}-(e^{-x/2}x)/2