What is (\partial)/(\partial x)(xe^{-ax^2}) ?

The answer to (\partial)/(\partial x)(xe^{-ax^2}) is e^{-ax^2}-2ae^{-ax^2}x^2

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(\partial)/(\partial x)(xe^{-ax^2})

\frac{\partial}{\partial x} (x e^{- a x^{2}})

e^{- a x^{2}} - 2 a e^{- a x^{2}} x^{2}

Solution steps

\frac{\partial}{\partial x} (x e^{- a x^{2}})

Apply the Product Rule:

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

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

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

\frac{\partial}{\partial x} (x) = 1

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

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

Simplify

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

= e^{- a x^{2}} - 2 a e^{- a x^{2}} x^{2}

\frac{d}{d x} (a (2 e^{- 3 x} x - 3 e^{- 3 x} x^{2}))

\frac{\partial}{\partial x} (- sin (4 x))

f (x) = x^{2} + 2 x + 3

\int 4 \frac{x}{( x + 2 ) ^{(\frac{1}{4})}} d x

x \to 1 - lim (3^{\frac{1}{x}})

What is (\partial)/(\partial x)(xe^{-ax^2}) ?
The answer to (\partial)/(\partial x)(xe^{-ax^2}) is e^{-ax^2}-2ae^{-ax^2}x^2