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Popular Calculus >

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

Solution

derivative of

ln (1 + e^{- a x})

Solution

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

Graph

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Frequently Asked Questions (FAQ)

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