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

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

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

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

- e^{- x y} x y^{2} + 3 e^{- x y} y

Solution steps

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

Treat

y

as a constant

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

= - e^{- x y} x y^{2} + 3 e^{- x y} y

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