What is the (xe^{-x^2})^' ?

The (xe^{-x^2})^' is e^{-x^2}-2e^{-x^2}x^2

What is the first (xe^{-x^2})^' ?

The first (xe^{-x^2})^' is e^{-x^2}-2e^{-x^2}x^2

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

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

Solution steps

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

Apply the Product Rule:

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

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

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

x^{'} = 1

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

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

Simplify

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

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

What is the (xe^{-x^2})^' ?
The (xe^{-x^2})^' is e^{-x^2}-2e^{-x^2}x^2
What is the first (xe^{-x^2})^' ?
The first (xe^{-x^2})^' is e^{-x^2}-2e^{-x^2}x^2