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

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

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

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

2 e^{- y} y - e^{- y} (x^{2} + y^{2})

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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x \to 0 lim (\frac{( 5 ^{x} - 7 ^{x} )}{x})

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