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

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

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(\partial)/(\partial x)(xye^{-x-y})

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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\int_{0}^{2} (2 - t) t d t

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