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

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

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

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

- e^{- \frac{x}{2}} x y + 2 e^{- \frac{x}{2}} y

Solution steps

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

Treat

y

as a constant

Take the constant out:

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

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

Apply the Product Rule:

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

f = e^{- \frac{x}{2}}, g = x

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

\frac{\partial}{\partial x} (e^{- \frac{x}{2}}) = - \frac{1}{2} e^{- \frac{x}{2}}

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

= 2 y ((- \frac{1}{2} e^{- \frac{x}{2}}) x + 1 \cdot e^{- \frac{x}{2}})

Simplify

2 y ((- \frac{1}{2} e^{- \frac{x}{2}}) x + 1 \cdot e^{- \frac{x}{2}}) : - e^{- \frac{x}{2}} x y + 2 e^{- \frac{x}{2}} y

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

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