What is the derivative of xe^{1/2 x} ?

The derivative of xe^{1/2 x} is (e^{x/2}x)/2+e^{x/2}

What is the first derivative of xe^{1/2 x} ?

The first derivative of xe^{1/2 x} is (e^{x/2}x)/2+e^{x/2}

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

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

Solution steps

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

Simplify

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

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

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

Simplify

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

\int \frac{x ^{6}}{x ^{14} - 3} d x

(0, 3), (1, 5)

x \to 6 lim (7)

\int \frac{sin ( x )}{( 3 + cos ( x ) ) ^{2}} d x

\int_{0}^{b} \frac{1}{4 - x ^{2}} d x

What is the derivative of xe^{1/2 x} ?
The derivative of xe^{1/2 x} is (e^{x/2}x)/2+e^{x/2}
What is the first derivative of xe^{1/2 x} ?
The first derivative of xe^{1/2 x} is (e^{x/2}x)/2+e^{x/2}