What is (\partial)/(\partial x)(ln(xe^{-6x})) ?

The answer to (\partial)/(\partial x)(ln(xe^{-6x})) is 1/x-6

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(\partial)/(\partial x)(ln(xe^{-6x}))

\frac{\partial}{\partial x} (ln (x e^{- 6 x}))

\frac{1}{x} - 6

Solution steps

\frac{\partial}{\partial x} (ln (x e^{- 6 x}))

Simplify

ln (x e^{- 6 x}) : ln (x) - 6 x

= \frac{\partial}{\partial x} (ln (x) - 6 x)

Apply the Sum/Difference Rule:

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

= \frac{\partial}{\partial x} (ln (x)) - \frac{\partial}{\partial x} (6 x)

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

\frac{\partial}{\partial x} (6 x) = 6

= \frac{1}{x} - 6

x \to + 0 lim ((\frac{1}{x ^{2}})^{2})

n \to - \infty lim (e^{n})

n = 0 \sum \infty e^{2 n}

x \to 0 lim (\frac{6 ^{x} - 2 ^{x}}{x})

x \to 0 + lim (x^{3} ln (x))

What is (\partial)/(\partial x)(ln(xe^{-6x})) ?
The answer to (\partial)/(\partial x)(ln(xe^{-6x})) is 1/x-6