What is the derivative of x^3e^{-x^2} ?

The derivative of x^3e^{-x^2} is 3e^{-x^2}x^2-2e^{-x^2}x^4

What is the first derivative of x^3e^{-x^2} ?

The first derivative of x^3e^{-x^2} is 3e^{-x^2}x^2-2e^{-x^2}x^4

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\frac{d}{d x} (x^{3} e^{- x^{2}})

3 e^{- x^{2}} x^{2} - 2 e^{- x^{2}} x^{4}

Solution steps

\frac{d}{d x} (x^{3} e^{- x^{2}})

Apply the Product Rule:

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

f = x^{3}, g = e^{- x^{2}}

= \frac{d}{d x} (x^{3}) e^{- x^{2}} + \frac{d}{d x} (e^{- x^{2}}) x^{3}

\frac{d}{d x} (x^{3}) = 3 x^{2}

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

= 3 x^{2} e^{- x^{2}} + (- 2 e^{- x^{2}} x) x^{3}

Simplify

3 x^{2} e^{- x^{2}} + (- 2 e^{- x^{2}} x) x^{3} : 3 e^{- x^{2}} x^{2} - 2 e^{- x^{2}} x^{4}

= 3 e^{- x^{2}} x^{2} - 2 e^{- x^{2}} x^{4}

\frac{d}{d x} ((5 x + 3)^{2})

y^{^{'}} + y = x e^{- x} + 1

\frac{d x}{d t} + 2 t x = 1

tan (4) x^{3}

x \to 0 lim (\frac{x}{x ^{2} + 7 x})

What is the derivative of x^3e^{-x^2} ?
The derivative of x^3e^{-x^2} is 3e^{-x^2}x^2-2e^{-x^2}x^4
What is the first derivative of x^3e^{-x^2} ?
The first derivative of x^3e^{-x^2} is 3e^{-x^2}x^2-2e^{-x^2}x^4