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

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

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

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

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

\frac{d}{d x} (x) = \frac{1}{2 x}

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

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

Simplify

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

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

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