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Popular >

derivative of 5x^3e^{-x}

Solution

\frac{d}{d x} (5 x^{3} e^{- x})

Solution

5 (3 e^{- x} x^{2} - e^{- x} x^{3})

Solution steps

\frac{d}{d x} (5 x^{3} e^{- x})

Take the constant out:

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

= 5 \frac{d}{d x} (x^{3} e^{- x})

Apply the Product Rule:

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

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

= 5 (\frac{d}{d x} (x^{3}) e^{- x} + \frac{d}{d x} (e^{- x}) x^{3})

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

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

= 5 (3 x^{2} e^{- x} + (- e^{- x}) x^{3})

Simplify

= 5 (3 e^{- x} x^{2} - e^{- x} x^{3})

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Frequently Asked Questions (FAQ)

What is the derivative of 5x^3e^{-x} ?
The derivative of 5x^3e^{-x} is 5(3e^{-x}x^2-e^{-x}x^3)
What is the first derivative of 5x^3e^{-x} ?
The first derivative of 5x^3e^{-x} is 5(3e^{-x}x^2-e^{-x}x^3)