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

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

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

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

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

Solution steps

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

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}

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

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

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

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

Simplify

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

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

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