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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

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

(3 x + 7) x^{2}

\frac{d}{d x} (- \frac{32}{x ^{3}} + \frac{4}{x ^{2}})

\int_{2}^{4} \frac{x ^{2} + 1}{x ^{2} - 5} d x

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

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