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

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

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(\partial)/(\partial x)(-xe^{-xy})

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

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

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

= - \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}

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

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

Simplify

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

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