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

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

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(\partial)/(\partial x)(A(e^{-x}-e^{-x}x))

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

A (e^{- x} x - 2 e^{- x})

Solution steps

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

Take the constant out:

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

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

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

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

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

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

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

Simplify

= A (e^{- x} x - 2 e^{- x})

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