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

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

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

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

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

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

Solution steps

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

Simplify

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

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