What is the derivative of ln((xe^{-2x}^{1/3})) ?

The derivative of ln((xe^{-2x}^{1/3})) is (1-2x)/(3x)

What is the first derivative of ln((xe^{-2x}^{1/3})) ?

The first derivative of ln((xe^{-2x}^{1/3})) is (1-2x)/(3x)

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\frac{d}{d x} (ln ((x e^{- 2 x})^{\frac{1}{3}}))

\frac{1 - 2 x}{3 x}

Solution steps

\frac{d}{d x} (ln ((x e^{- 2 x})^{\frac{1}{3}}))

Apply the chain rule:

\frac{1}{( x e ^{- 2 x} ) ^{\frac{1}{3}}} \frac{d}{d x} ((x e^{- 2 x})^{\frac{1}{3}})

= \frac{1}{( x e ^{- 2 x} ) ^{\frac{1}{3}}} \frac{d}{d x} ((x e^{- 2 x})^{\frac{1}{3}})

\frac{d}{d x} ((x e^{- 2 x})^{\frac{1}{3}}) = \frac{1 - 2 x}{3 e ^{\frac{2 x}{3}} x ^{\frac{2}{3}}}

= \frac{1}{( x e ^{- 2 x} ) ^{\frac{1}{3}}} \cdot \frac{1 - 2 x}{3 e ^{\frac{2 x}{3}} x ^{\frac{2}{3}}}

Simplify

\frac{1}{( x e ^{- 2 x} ) ^{\frac{1}{3}}} \cdot \frac{1 - 2 x}{3 e ^{\frac{2 x}{3}} x ^{\frac{2}{3}}} : \frac{1 - 2 x}{3 x}

= \frac{1 - 2 x}{3 x}

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n = 1 \sum \infty \frac{4}{3 ^{n}}

\int_{6}^{8} \frac{8}{( x - 6 ) ^{3}} d x

What is the derivative of ln((xe^{-2x}^{1/3})) ?
The derivative of ln((xe^{-2x}^{1/3})) is (1-2x)/(3x)
What is the first derivative of ln((xe^{-2x}^{1/3})) ?
The first derivative of ln((xe^{-2x}^{1/3})) is (1-2x)/(3x)