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

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

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

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

e^{- x^{2} - y^{2}} - 2 e^{- x^{2} - y^{2}} x^{2}

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

x \to 0 lim (\frac{e ^{x} - 1}{9 x})

\frac{d y}{d x} = 6 x^{2} + 3

\int_{0}^{1} ln (6 x) d x

\int_{0}^{4} \frac{x ^{2}}{x ^{3} + 7} d x

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

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