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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derivative of

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}}) = e^{\frac{x}{2}} \frac{1}{2}

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

Simplify

1 \cdot e^{\frac{x}{2}} + e^{\frac{x}{2}} \frac{1}{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