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

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

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

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

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

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

Solution steps

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

Apply exponent rule:

\frac{1}{a} = a^{- 1}

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