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Popular >

derivative of ln(1+e^{-x})

Solution

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

Solution

- \frac{e ^{- x}}{1 + e ^{- x}}

Solution steps

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

Apply the chain rule:

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

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

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

= \frac{1}{1 + e ^{- x}} (- e^{- x})

Simplify

\frac{1}{1 + e ^{- x}} (- e^{- x}) : - \frac{e ^{- x}}{1 + e ^{- x}}

= - \frac{e ^{- x}}{1 + e ^{- x}}

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Frequently Asked Questions (FAQ)

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