What is (\partial)/(\partial x)((3-xy)e^{-xy}) ?

The answer to (\partial)/(\partial x)((3-xy)e^{-xy}) is -4e^{-xy}y+e^{-xy}xy^2

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(\partial)/(\partial x)((3-xy)e^{-xy})

\frac{\partial}{\partial x} ((3 - x y) e^{- x y})

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

Solution steps

\frac{\partial}{\partial x} ((3 - x y) e^{- x y})

Treat

y

as a constant

Apply the Product Rule:

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

f = 3 - x y, g = e^{- x y}

= \frac{\partial}{\partial x} (3 - x y) e^{- x y} + \frac{\partial}{\partial x} (e^{- x y}) (3 - x y)

\frac{\partial}{\partial x} (3 - x y) = - y

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

= (- y) e^{- x y} + (- y e^{- y x}) (3 - x y)

Simplify

(- y) e^{- x y} + (- y e^{- y x}) (3 - x y) : - 4 e^{- x y} y + e^{- x y} x y^{2}

= - 4 e^{- x y} y + e^{- x y} x y^{2}

What is (\partial)/(\partial x)((3-xy)e^{-xy}) ?
The answer to (\partial)/(\partial x)((3-xy)e^{-xy}) is -4e^{-xy}y+e^{-xy}xy^2