What is the derivative of e^{-x}(x-1) ?

The derivative of e^{-x}(x-1) is -e^{-x}x+2e^{-x}

What is the first derivative of e^{-x}(x-1) ?

The first derivative of e^{-x}(x-1) is -e^{-x}x+2e^{-x}

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\frac{d}{d x} (e^{- x} (x - 1))

- e^{- x} x + 2 e^{- x}

Solution steps

\frac{d}{d x} (e^{- x} (x - 1))

Apply the Product Rule:

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

f = e^{- x}, g = x - 1

= \frac{d}{d x} (e^{- x}) (x - 1) + \frac{d}{d x} (x - 1) e^{- x}

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

\frac{d}{d x} (x - 1) = 1

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

Simplify

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

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

What is the derivative of e^{-x}(x-1) ?
The derivative of e^{-x}(x-1) is -e^{-x}x+2e^{-x}
What is the first derivative of e^{-x}(x-1) ?
The first derivative of e^{-x}(x-1) is -e^{-x}x+2e^{-x}