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

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

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

\frac{d}{d x} (- 4 sin (x) cos (x))

\int \frac{3}{x + 4} d x

f (x) = x^{3} + x, at x = 2

cos (sin^{2} (x - 2))

x \to 0 + lim (arctan (ln (0)))

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