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

The derivative of (1+e^{-x}^{-1}) is (e^{-x})/((1+e^{-x))^2}

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

The first derivative of (1+e^{-x}^{-1}) is (e^{-x})/((1+e^{-x))^2}

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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

Solution steps

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

What is the derivative of (1+e^{-x}^{-1}) ?
The derivative of (1+e^{-x}^{-1}) is (e^{-x})/((1+e^{-x))^2}
What is the first derivative of (1+e^{-x}^{-1}) ?
The first derivative of (1+e^{-x}^{-1}) is (e^{-x})/((1+e^{-x))^2}