What is expand e^x(-xe^{-x}-e^{-x}) ?

The solution to expand e^x(-xe^{-x}-e^{-x}) is -x-1

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expand e^x(-xe^{-x}-e^{-x})

expand

e^{x} (- x e^{- x} - e^{- x})

- x - 1

Solution steps

e^{x} (- x e^{- x} - e^{- x})

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

Apply the distributive law:

a (b - c) = ab - a c

a = e^{x}, b = - x e^{- x}, c = e^{- x}

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

Apply minus-plus rules

+ (- a) = - a

= - e^{x} e^{- x} x - e^{x} e^{- x}

Simplify

- e^{x} e^{- x} x - e^{x} e^{- x} : - x - 1

= - x - 1

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What is expand e^x(-xe^{-x}-e^{-x}) ?
The solution to expand e^x(-xe^{-x}-e^{-x}) is -x-1