What is the derivative of ln(xe^{-5x}) ?

The derivative of ln(xe^{-5x}) is 1/x-5

What is the first derivative of ln(xe^{-5x}) ?

The first derivative of ln(xe^{-5x}) is 1/x-5

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

\frac{1}{x} - 5

Solution steps

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

Simplify

ln (x e^{- 5 x}) : ln (x) - 5 x

= \frac{d}{d x} (ln (x) - 5 x)

Apply the Sum/Difference Rule:

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

= \frac{d}{d x} (ln (x)) - \frac{d}{d x} (5 x)

\frac{d}{d x} (ln (x)) = \frac{1}{x}

\frac{d}{d x} (5 x) = 5

= \frac{1}{x} - 5

\int_{- 2}^{2} (\frac{1}{( x + 4 ) ( x - 4 )}) d x

x \to 0 - lim (csc (x) (x - 1))

\int_{- \infty}^{0} x^{2} e^{x} d x

\frac{\partial}{\partial x} (\frac{e ^{x z + y}}{z ^{2} + w})

\frac{d}{d x} (\frac{1}{( x + 9 ) ^{2}})

What is the derivative of ln(xe^{-5x}) ?
The derivative of ln(xe^{-5x}) is 1/x-5
What is the first derivative of ln(xe^{-5x}) ?
The first derivative of ln(xe^{-5x}) is 1/x-5