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Popular >

derivative of y=xe^{-x^2}

Solution

derivative of

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

Solution

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

Solution steps

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

Apply the Product Rule:

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

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

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

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

\frac{d}{d x} (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}

Popular Examples

integral of (x+3)/(x-2)

\int \frac{x + 3}{x - 2} d x

derivative of (t^2)/(t-2)derivative of

\frac{t ^{2}}{t - 2}

integral of (6x^5+7x)

\int (6 x^{5} + 7 x) d x

integral of xsqrt(-x^2+x)

\int x - x^{2} + x d x

(dy)/(dx)=(y^2)/(e^x),y(0)=-2

\frac{d y}{d x} = \frac{y ^{2}}{e ^{x}}, y (0) = - 2

Frequently Asked Questions (FAQ)

What is the derivative of y=xe^{-x^2} ?
The derivative of y=xe^{-x^2} is e^{-x^2}-2e^{-x^2}x^2
What is the first derivative of y=xe^{-x^2} ?
The first derivative of y=xe^{-x^2} is e^{-x^2}-2e^{-x^2}x^2