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

The derivative of a(2e^{-5x}x-5e^{-5x}x^2) is a(25e^{-5x}x^2-20e^{-5x}x+2e^{-5x})

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

The first derivative of a(2e^{-5x}x-5e^{-5x}x^2) is a(25e^{-5x}x^2-20e^{-5x}x+2e^{-5x})

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\frac{d}{d x} (a (2 e^{- 5 x} x - 5 e^{- 5 x} x^{2}))

a (25 e^{- 5 x} x^{2} - 20 e^{- 5 x} x + 2 e^{- 5 x})

Solution steps

\frac{d}{d x} (a (2 e^{- 5 x} x - 5 e^{- 5 x} x^{2}))

Take the constant out:

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

= a \frac{d}{d x} (2 e^{- 5 x} x - 5 e^{- 5 x} x^{2})

Apply the Sum/Difference Rule:

(f \pm g)^{'} = f^{'} \pm g^{'}

= a (\frac{d}{d x} (2 e^{- 5 x} x) - \frac{d}{d x} (5 e^{- 5 x} x^{2}))

\frac{d}{d x} (2 e^{- 5 x} x) = 2 (- 5 e^{- 5 x} x + e^{- 5 x})

\frac{d}{d x} (5 e^{- 5 x} x^{2}) = 5 (- 5 e^{- 5 x} x^{2} + 2 e^{- 5 x} x)

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

Expand

2 (- 5 e^{- 5 x} x + e^{- 5 x}) - 5 (- 5 e^{- 5 x} x^{2} + 2 e^{- 5 x} x) : 25 e^{- 5 x} x^{2} - 20 e^{- 5 x} x + 2 e^{- 5 x}

= a (25 e^{- 5 x} x^{2} - 20 e^{- 5 x} x + 2 e^{- 5 x})

What is the derivative of a(2e^{-5x}x-5e^{-5x}x^2) ?
The derivative of a(2e^{-5x}x-5e^{-5x}x^2) is a(25e^{-5x}x^2-20e^{-5x}x+2e^{-5x})
What is the first derivative of a(2e^{-5x}x-5e^{-5x}x^2) ?
The first derivative of a(2e^{-5x}x-5e^{-5x}x^2) is a(25e^{-5x}x^2-20e^{-5x}x+2e^{-5x})