What is the derivative of 1/2 x^2e^{-2x} ?

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

What is the first derivative of 1/2 x^2e^{-2x} ?

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

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

e^{- 2 x} x (- x + 1)

Solution steps

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

Take the constant out:

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

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

Apply the Product Rule:

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

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

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

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

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

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

Simplify

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

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

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