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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

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