What is the derivative of x*e^{-2x} ?

The derivative of x*e^{-2x} is e^{-2x}-2e^{-2x}x

What is the first derivative of x*e^{-2x} ?

The first derivative of x*e^{-2x} is e^{-2x}-2e^{-2x}x

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\frac{d}{d x} (x \cdot e^{- 2 x})

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

Solution steps

\frac{d}{d x} (x e^{- 2 x})

Apply the Product Rule:

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

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

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

\frac{d x}{d x} = 1

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

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

Simplify

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

What is the derivative of x*e^{-2x} ?
The derivative of x*e^{-2x} is e^{-2x}-2e^{-2x}x
What is the first derivative of x*e^{-2x} ?
The first derivative of x*e^{-2x} is e^{-2x}-2e^{-2x}x