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

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

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

\frac{\partial}{\partial x} (x \cdot 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}

\int x 2 - x^{2} d x

\frac{x ^{2} d y}{d x} = y - x y, y [- 1] = - 4

\frac{\partial}{\partial x} (\frac{sin ( x )}{y ^{2}})

\frac{\partial}{\partial u ( v )} (\frac{e ^{v}}{u ( v ) + v ^{7}})

\int \frac{1}{x ^{2} - 169} d x

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