What is the derivative of x^3e^{-8x} ?

The derivative of x^3e^{-8x} is 3e^{-8x}x^2-8e^{-8x}x^3

What is the first derivative of x^3e^{-8x} ?

The first derivative of x^3e^{-8x} is 3e^{-8x}x^2-8e^{-8x}x^3

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\frac{d}{d x} (x^{3} e^{- 8 x})

3 e^{- 8 x} x^{2} - 8 e^{- 8 x} x^{3}

Solution steps

\frac{d}{d x} (x^{3} e^{- 8 x})

Apply the Product Rule:

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

f = x^{3}, g = e^{- 8 x}

= \frac{d}{d x} (x^{3}) e^{- 8 x} + \frac{d}{d x} (e^{- 8 x}) x^{3}

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

\frac{d}{d x} (e^{- 8 x}) = - 8 e^{- 8 x}

= 3 x^{2} e^{- 8 x} + (- 8 e^{- 8 x}) x^{3}

Simplify

= 3 e^{- 8 x} x^{2} - 8 e^{- 8 x} x^{3}

What is the derivative of x^3e^{-8x} ?
The derivative of x^3e^{-8x} is 3e^{-8x}x^2-8e^{-8x}x^3
What is the first derivative of x^3e^{-8x} ?
The first derivative of x^3e^{-8x} is 3e^{-8x}x^2-8e^{-8x}x^3