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

The derivative of a(e^{-5x}-5e^{-5x}x) is a(25e^{-5x}x-10e^{-5x})

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

The first derivative of a(e^{-5x}-5e^{-5x}x) is a(25e^{-5x}x-10e^{-5x})

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

a (25 e^{- 5 x} x - 10 e^{- 5 x})

Solution steps

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

Take the constant out:

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

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

Apply the Sum/Difference Rule:

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

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

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

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

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

Expand

- 5 e^{- 5 x} - 5 (- 5 e^{- 5 x} x + e^{- 5 x}) : 25 e^{- 5 x} x - 10 e^{- 5 x}

= a (25 e^{- 5 x} x - 10 e^{- 5 x})

What is the derivative of a(e^{-5x}-5e^{-5x}x) ?
The derivative of a(e^{-5x}-5e^{-5x}x) is a(25e^{-5x}x-10e^{-5x})
What is the first derivative of a(e^{-5x}-5e^{-5x}x) ?
The first derivative of a(e^{-5x}-5e^{-5x}x) is a(25e^{-5x}x-10e^{-5x})